PARTIALLY ORDERED VARIETIES AND QUASIVARIETIES 1

PARTIALLY ORDERED VARIETIES AND QUASIVARIETIES

DON PIGOZZI

Abstract. The recent development of abstract algebraic logic has led to a reconsiderationof the universal algebraic theory of ordered algebras from a new perspective. The familiarequational logic of Birkhoff can be naturally viewed as one of the deductive systems thatconstitute the main object of study in abstract algebraic logic; technically it is a deductivesystem of dimension two. Large parts of universal algebra turn out to fit smoothly withinthe matrix semantics of this deductive system. Ordered algebras in turn can be viewed asspecial kinds of matrix models of other, closely related, 2-dimensional deductive systems.We consider here a more general notion of ordered algebra in which some operations maybe anti-monotone in some arguments; this leads to many different ordered equational logicsover the same language. These lectures will be a introduction to universal ordered algebrafrom this new perspective.

1. Introduction

Algebraic logic, in particular abstract algebraic logic (AAL) as presented for example inthe monograph [1], can be viewed as the study of logical equivalence, or more precisely asthe study of properties of logical propositions upon abstraction from logical equivalence.Traditional algebraic logic concentrates on the algebraic structures that result from thisprocess, while AAL is more concerned with the process of abstraction itself. These lecturesare the outgrowth of an attempt to apply the methods of AAL to logical implication inplace of logical equivalence, with special emphasis on those circumstances in which logicalimplication cannot be expressed in terms of logical equivalence. This persepective resultsin a theory of ordered algebraic systems markedly different from the traditional one aspresented for example in [14], but of course the overlap is considerable.

Underlying the development is the conception of order algebras as the reduced matrixsemantics of inequational logic, an alternative to Birkhoff’s equational logic. However themetalogical roots are] mostly kept in the background in order to simplify the expositionand avoid distracting the reader from the central part of the theory. We do however includeparenthetical remarks explicitly marking connections with AAL when appropriate.

Some results can be found in the literature in a somewhat weaker form; for example, theorder H-S-P theorem (Theorem 3.14) was obtained by Bloom [4] in 1976 under the conditionthat the fundamental operations of the algebra are monotone at all argument positions. Onthe other hand, most of the basic algebraic theory of partially ordered algebras presentedin Section 2 can be found in a more general form in the work of A. I. Mal’cev, and that ofhis scientific progeny, on quasivarieties; a good appreciation of this work can be obtained

Date: January 11, 2004.Revised notes of lectures based on joint work with Katarzyna Pa lasinska given at the CAUL, Lisbon in

September of 2003, and at the Universidad Catolica, Santiago in November of 2003.

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2 DON PIGOZZI

from the monographs [15, 23]. In the Russian literature the term “quasivariety” can refer toan arbitrary strict universal Horn class, and consequently a quasivariety can mean a classof structures with relations as well as operation. The S-L-P theorem (Theorem 3.17) canbe obtained from Mal’cev’s well-known algebraic characterization of strict universal Hornclasses [22]; the particular form of Theorem 3.17 can be found in [17]. But the Russian workalways includes equality as a fundamental predicate, while inequational logic is a specialkind of universal Horn logic without equality (UHLWE)—one of the so-called equivalentiallogics. Equivalential UHLWE’s, and a more general class, the protoalgebraic UHLWE’s,constitute a family of logical systems whose reduced model theory is most like algebra.These systems have been extensively investigated in the literature of AAL [2, 7, 8, 12, 13];the papers [2, 12] contain general forms of the H-S-P theorem from which Theorem 3.14 canbe easily extracted. We also call attention to the earlier papers on equivalential sententiallogics [5, 24], and to the monograph [6], which is a comprehensive resource for the theoryof protoalgebraic and equivalential sentential logics.

In spite of the above remarks, we feel there is value in developing the algebraic theory oforder algebras from first principles and in parallel with that of ordinary, unordered, algebrasrather than simply as an adjunction to the latter theory. We also think it is also importantthat the theory be purely algebraic rather than grafted on to the the theory of equivalentialUHLWE’s.

Fundamental for our work is the notion of a signature Σ augmented by a polarity, anarbitrary assignment of positive or negative polarity to each argument position of eachoperation symbol of Σ specifying whether the operation is to be monotone or antimonotoneat that position. For example, in a Heyting algebra 〈A,+, ·,→, 0, 1〉 with the natural order,under the natural polarity + and · are each positive in both arguments and the implication→ is positive in the second and negative in the first. Admitting signatures with arbitrarypolarity greatly expands the scope of the ordered algebras to which the theory applies andis one of the major innovations. A similarly broad approach can also be found in the workof Dunn [11] and is a natural development of his metalogical investigations of relevanceand other so-called substructural logics. His theory of gaggles [9, 10] he presents orderedalgebra models for a very wide class of substructural logics, including linear logic, in whichthe polarity of the various operations play a key role.

1.1. Outline of Lectures. In the first section we introduce the notion of a polarity ρ ona signature Σ, and of a ρ-partially ordered algebra A, consisting of a Σ-algebra A and apartial ordering ≤A such that the fundamental operations of A are monotone or antimono-tone in each argument in accordance with ρ. Order homomorphisms of ρ-poalgebras arehomomorphisms of the underlying algebras that preserve the designated partial orders. ρ-quasi-orders of A are quasi-orderings of the universe of the underlying algebra that includethe designated partial ordering ≤A and respect the polarity in the same sense. They playthe role in the theory of ρ-poalgebras that congruences play in universal algebra, and aresimilarly important to the theory. Quotients of ρ-poalgebras are taken with respect to ρ-quasi-orders, and order analogues of the homomorphism, isomorphism, and correspondencetheorems of universal algebra are formulated and proved. Order subalgebras, direct prod-ucts, reduced and ultraproduces, and direct limits of ρ-poalgebras are defined. Subdirect

January 11, 2004

PARTIALLY ORDERED VARIETIES AND QUASIVARIETIES 3

products are defined and an analogue of Birkhoff’s representation as a subdirect productof subdirectly irreducible ρ-poalgebras is proved.

Inequational logic is developed in Section 3. A quasi-inidentity is a strict universalHorn formula whose atomic formulas is an inequality between terms; the special case of auniversally quantified inequality is an inidentity. The class of all ρ-poalgebras that satisfysome set of quasi-inidentities is called a quasi-povariety ; if all the quasi-inidentities areinidentities, it is called a povariety. The notion of a freely generated ρ-algebra over aclass of ρ-poalgebras is discussed, and it is used in proving the order H-S-P theorem, i.e.,that a class of ρ-poalgebra is an ρ-povariety iff it is closed under the formation of orderhomomorphic images, subalgebras, and direct products. We prove a analogous operator-theoretic characterization of quasi-povarieties as the class of ρ-poalgebras closed under ordersubalgebras, direct products, and direct limits.

Several examples of ρ-povarieties are given: the povariety of lattices as poalgebras and theρ-povariety of partially ordered left residuated monoids (POLRM). Partially ordered groupsform a sub-povariety of POLRM. Lattices are more commonly viewed as algebras ratherthan poalgebras, and in this conception are defined by identities rather than indentities.POLRMs cannot be represented as algebras in this way. This special property of thepovariety of lattices, algebraizability, is investigated in Section 4. It is a special case ofthe well-known notion from AAL of the same name. In the last section we investigate thegeneration of ρ-quasi-orders. We prove a Mal’cev-like combinatorial lemma characterizingthe ρ-quasi-ordering generated by an arbitrary set of pairs of elements on a ρ-poalgebrain terms of polynomials. This is used to obtain two characterization of those ρ-povarietieswith permuting ρ-quasi-orders—one in terms of the existence of certain quasi-indentitiesand another, related, one in terms of inidentities with additional conditions on the polarityof the terms comprising the inidentities. The latter characterization is used to show thatthe ρ-quasi-orders of POLRM’s permute.

POLRM’s together with the various other ρ-povarieties related to substructual logics canbe regarded as the paradigms ρ-povarieties from our perspective. The algebraizable ones,for example the partially ordered commutative residuated integral monoids (POLCRIM’s)have been extensively investigated using the methods of standard universal algebra; thepapers [3, 19, 20, 21, 25, 26] are representative of the publications in the area. We havesome hope that the methods of ordered universal algebra will prove to be useful in theinvestigation of POLRM’s and other non-algebraizable ρ-povarieties.

2. Basic Algebraic Theory

A reflexive and transitive relation on a set is called a quasi-ordering (a qordering). Thus≤ ⊆ A2 is a qordering if

• (∀ a ∈ A)(a ≤ a),• (∀ a, b, c ∈ A)(a ≤ b and b ≤ c =⇒ a ≤ c).

It is a partial ordering (a pordering) if it is also antisymmetric:• (∀ a, b ∈ A)(a ≤ b and b ≤ a =⇒ a = b).

〈A,≤〉 is called a quasi-ordered set (a qoset)—a partially ordered set (a poset) if ≤ is apordering. The reverse of ≤ is denoted by ≥ or ≤−1. Thus a ≥ b iff b ≤ a.

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4 DON PIGOZZI

A signature or language type is a set Σ together with an order or arity function o:Σ → ω(ω is the set of natural numbers). o(σ) is the order or arity of σ ∈ Σ and specifies thenumber of arguments that σ takes. Σn = {σ ∈ Σ : o(σ) = n }. A Σ-algebra is a systemA = 〈A, σA〉σ∈Σ such that A is a nonempty set and σA is an operation on A of order o(σ),i.e., σA:Ao(σ) → A.

A polarity for Σ is a fixed but arbitrary assignment of a polarity, either positive ornegative, to each argument position of each operation symbol in Σ. If, as commonly done,we identify a natural number with the set of natural numbers less than it, we can define apolarity to be a bfunction

ρ:( ⋃σ∈Σ

σ × o(σ))→ {+,−}.

σ is said to be of positive or negative polarity at the i-th argument (with respect to ρ) ifρ(σ, i) is + or −, respectively. With slight abuse of notation we take ρ+(σ) and ρ−(σ) to bethe sets of arguments of σ of positive and negative polarity, respectively. So ρ+(σ) = { i <o(σ) : ρ(σ, i) = + } and ρ−(σ) = { i < o(σ) : ρ(σ, i) = −}. We also say that σ is monotoneor antimonotone in the i-th argument if ρ(σ, i) = + or ρ(σ, i) = −, respectively.

Definition 2.1. Let Σ be a signature and ρ a polarity for Σ. Let A be a Σ-algebra. Aquasi-ordering ≤ on the universe A of A is said to be a ρ-quasi-ordering, a ρ-qordering, ofA if, for every σ ∈ Σn and all a0, . . . , an−1, b0, . . . , bn−1 ∈ A,

(∀ i ∈ ρ+(σ))(ai ≤ bi)) and ∀ j ∈ ρ−(σ))(aj ≥ bj))

=⇒ σA(a0, . . . , an−1) ≤ σA(b0, . . . , bn−1).

This condition is called ρ-tonicity. A pordering of A that satisfies it is a ρ-partial ordering, aρ-pordering, of A, and the pair 〈A,≤〉 is called a ρ-partially ordered Σ-algebra, a ρ-poalgebrafor short.

We note for future reference that, in the presence of transitivity, to verify the ρ-tonicitycondition it suffices to show that, for each σ ∈ Σn, each i < n, and all a, b, c0, . . . , ci−1,ci+1, . . . , cn−1 ∈ A, if i ∈ ρ+(σ), then

a ≤ b =⇒ σA(c0, . . . , ci−1, a, ci+1, . . . , cn−1) ≤ σA(c0, . . . , ci−1, b, ci+1, . . . , cn−1),

and if i ∈ ρ−(σ), then

a ≥ b =⇒ σA(c0, . . . , ci−1, a, ci+1, . . . , cn−1) ≤ σA(c0, . . . , ci−1, b, ci+1, . . . , cn−1)

The polarity ρ such that ρ(σ, i) = + for every σ ∈ Σ and every i < o(σ) is said to becompletely positive or completely monotone. In this case ρ-poalgebras are called simplypoalgebras.

The group of integers with the natural order⟨〈Z,+〉,≤

⟩and the semiring of natural

numbers⟨〈ω, +, ·〉,≤

⟩are poalgebras over the signatures {+} and {+, ·} respectively.

The group of integers in the form⟨〈Z,+,−〉,≤

⟩is not a poalgebra over the signature

{+,−} because n ≤ m implies −n ≥ −m. It is however a ρ-poalgebra where ρ(+, 0) =ρ(−, 1) = + and ρ(−, 0) = −. Any Boolean algebra and more generally any Heyting algebra⟨〈A,+, ·,−,→, 0, 1〉,≤

⟩with the natural order given by a ≤ b iff a → b = 1 is a ρ-poalgebra

where ρ(+, 0) = ρ(+, 1) = ρ(·, 0) = ρ(·, 1) = ρ(→, 1) = + and ρ(−, 0) = ρ(→, 0) = −.

January 11, 2004


These are special cases of a more general class of ordered algebras that we want toconsider. A partially ordered left-residuated monoid, a POLRM is a structure

⟨〈A, ·,→

, 1〉,≤⟩, where• 〈A, ·, 1〉 is a monoid (i.e., · is an associative binary operation on A and 1·a = a·1 = a

for all a ∈ A).• · is monotone in both arguments, i.e., a ≤ b implies a · c ≤ b · c and c · a ≤ c · b,

for all a, b, c ∈ A. In terms of the terminology introduced above this just says that⟨〈A, ·〉,≤

⟩is a poalgebra.

• For all a, b ∈ A, a → b is the largest element z of A such that z · a ≤ b, i.e.,

(∀ z ∈ A)(z · a ≤ b ⇐⇒ z ≤ a → b).

This last condition is called the left residuation condition.

Proposition 2.2. Every left-residuated ordered monoid is a ρ-poalgebra where ρ(·, 0) =ρ(·, 1) = ρ(→, 1) = + and ρ(→, 0) = −.

Proof. . · is monotone in both arguments by definition. To verify → is monotone in thesecond and anti-monotone in the first note that, from x → y ≤ x → y we get(

∀x, y)(

x · (x → y) ≤ y)

by residuation. This condition is called detachment because of its strong resemblance tothe detachment rule of logic.

If a ≤ b, then c·(c → a) ≤ a ≤ b by detachment, and hence c → a ≤ c → b by residuation.If a ≥ b, then b · (a → c) ≤ a · (a → c) ≤ c by detachment and the monotonicity of · in firstargument. Thus a → c ≤ b → c by residuation. �

The identity relation on a set A is denoted by ∆A. ∆A is a ρ-pordering of A for everyΣ-algebra A and every polarity ρ. ∆Zn is the only pordering of the additive group ofintegers modulo n for any finite nonzero n. If ≤ is a ρ-pordering of A, then so is its reverse≥. Thus if there is a ρ-ordering of A different from the identity, then there are at leastthree distinct ρ-algebras with the the same underlying algebra A (under the assumption ofcourse that A is non-trivial).

In the sequel explicit reference to the signature of an algebra is usually omitted whenthe signature is generic and all algebras are of the same signature. Also in the sequelρ-poalgebras will be represented by boldface calligraphic letters with the correspondingboldface Roman letters for the underlying algebras. Thus we have A = 〈A,≤A〉, B =〈B,≤A〉, etc. The superscript is necessary to distinguish between different ρ-porderings ofthe same algebra; for example 〈Z,∆Z〉, 〈Z,≤〉, and 〈Z,≥〉 (with ≤ the natural ordering)are different poalgebras with the same underlying algebra. However, when no confusion islikely we normally omit the superscript.

Let ≤ be a ρ-qordering of an algebra A. A congruence relation α on A is said to becompatible with ≤ if a ≤ b implies a′ ≤ b′ for all a′, b′ such that aα a′ and b α b′. It is easyto check that α is compatible with ≤ iff α ⊆ ≤. If α is compatible with ≤ we define thequotient ≤/α on the quotient set A/α by the condition

[a]α ≤/α [b]α if a ≤ b,

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6 DON PIGOZZI

where [a]α is the α-equivalence class of a; note that compatibility insures that ≤/α is well-defined in the sense that it does not depend on the choice of the representatives a of [a]αand b of [b]α.

Proposition 2.3. Let ≤ be a ρ-qordering of an algebra A and let α be a congruence on Athat is compatible with ≤. Then ≤/α is a ρ-qordering of A/α. �

Let ≤≥= ≤ ∩ ≥; then a ≤≥ b iff a ≤ b and b ≤ a. For every qordering of a set A, ≤≥is an equivalence relation on A, and if ≤ is a ρ-qordering of A, then ≤≥ is a congruencerelation on A, and in fact it is the largest congruence relation compatible with ≤.1 Thusthe only congruence relation compatible with a pordering is the identity congruence. ≤≥is called the symmetrization of ≤.

Definition 2.4. Let A = 〈A,≤A〉 and B = 〈B,≤A〉 be ρ-poalgebras. A homomorphismh:A → B is an order homomorphism if h(≤A) ⊆ ≤B, i.e., a ≤A a′, implies h(a) ≤B h(a′)for all a, a′ ∈ A; in symbols h:A → B. The set of all order homomorphisms from A to Bis denoted by Hom(A,B).

h is an order monomorphism if it is a monomorphism (i.e., an injective homomorphism)of the underlying algebras and h(≤A) = ≤B ∩ B2. h is an order epimorphism if it is anepimorphism (i.e., a surjective homomorphism) of the underlying algebras. Finally, h isan order isomorphism if it is both an order monomorphism and an order epimorphism, insymbols h:A ∼= B. It is easy to see that an order homomorphism h is an order-isomorphismiff it is an isomorphism of the underlying algebras and h−1 is an order-homomorphismbetween B and A.

Note that an algebra homomorphism h:A → B is an order homomorphism iff h−1(≤B)⊇ ≤A. Thus an algebra isomorphism h:A ∼= B is an order isomorphism iff h−1(≤A) = ≤B.Let h:A → B be an algebra homomorphism. The congruence relation h−1(∆B) = { 〈a, a′〉 :h(a) = h(a′) } is called the kernel of h and is denoted by ker(h).

The class of ρ-poalgebras together with the ordered homomorphisms constitute a categoryin the natural sense.

Lemma 2.5. Let h:A → B be an order homomorphism. Then h−1(≤B) is a ρ-qorderingof A such that ≤A ⊆ h−1(≤B). Furthermore ker h = h−1(≤B) ∩ h−1(≥B).

Proof. Let ≤= h−1(≤B). The verification that ≤ is a quasi-ordering is straightforward.Assume ai ≤ a′i for all i ∈ ρ+(σ) and aj ≥ a′j for all j ∈ ρ−(σ). Then h(ai) ≤B h(a′)i forall i ∈ ρ+(σ) and h(aj) ≥B h(a′j) for all j ∈ ρ−(σ). Thus

h(σA(a0, . . . , an−1)) = σB(h(a0), . . . , h((an−1))

≤B σB(h(a′0), . . . , h((a′n−1)) = h(σA(a0, . . . , an−1)).

So σA(a0, . . . , an−1) h−1(≤B) σA(a0, . . . , an−1)). Hence h−1(≤B) is a ρ-qordering of A.Finally, we have h−1(≤B) ∩ h−1(≥B) = h−1(≤B ∩ ≥B) = h−1(∆B) = ker(h). �

Definition 2.6. Let h:A → B be an order homomorphism of ρ-poalgebras. h−1(≤B) iscalled the order-kernel of h, in symbols ordker(h).

1This is called the Leibniz congruence of ≤ in the terminology of AAL. See Remarks 2.18 and 3.5 below.

January 11, 2004


Proposition 2.7. An order homomorphism h:A → B is an order monomorphism iff itsorder kernel is ≤A.

Proof. ⇐= Suppose h−1(≤B) =≤A. Then ker(h) = h−1(≤B) ∩ h−1(≥B) =≤A ∩ ≤A=∆A. So h:A → B is an algebra monomorphism. ≤B ∩h(A)2 = hh−1(≤B) = h(≤A).

=⇒ Suppose h is an order monomorphism, i.e., ker(h) = ∆A and h(≤A) =≤B ∩h(A)2.Then h is algebra monomorphism and hh−1(≤B) =≤B ∩h(A)2 = h(≤A). Henceh−1(≤B) = ≤A since h is injective. �

Definition 2.8. Let A be a ρ-poalgebra. By a ρ-quasi-order, a ρ-qorder, of A we meana ρ-qordering ≤ of A, the underlying algebra of A, such that ≤A ⊆ ≤. The set of allρ-qorders of A is denoted by Qordρ(A).

We will use lower case Greek letters α, β, γ, . . . to represent ρ-qorders of A as well ascongruence relations on A. If α represents the ρ-qorder ≤, then the reverse ordering ≥ willbe represented by α−1.

Definition 2.9. Let A be a ρ-poalgebra and α ∈ Qordρ(A). The ρ-poalgebra 〈A/α ∩α−1, α/α ∩ α−1〉 is denoted by A/α and is called the quotient of A by α.

Theorem 2.10 (Order Homomorphism Theorem). Let A be a ρ-poalgebra and α ∈ Qordρ(A).(i) The natural map n:A → A/α ∩ α−1 is an order epimorphism from A onto A/α

with order kernel α.(ii) Let h:A → B be an order homomorphism such that α ⊆ h−1(≤B). Then there

exists a unique order homomorphism g:A/α → B such that h = g ◦ n. Moreover,for every a ∈ A, g([a]α∩α−1) = h(a).

Proof. (i). a ≤A a′ =⇒ aα a′ ⇐⇒ [a]α∩α−1 α/α ∩ α−1 [a′]α∩α−1 . But [a]α∩α−1 = n(a)and [a′]α∩α−1 = n(a′). n−1(α/α ∩ α−1) = α.

(ii). By assumption, α ⊆ h−1(≤B). So α ∩ α−1 ⊆ h−1(≤B) ∩ h−1(≥B) = ker(h). Soby the unordered homomorphism theorem, there is a unique g:A/α ∩ α−1 → B such thath = g ◦ n, and g([a]α∩α−1) = h(a) for each a ∈ A. It remains only to show that g is anorder homomorphism. [a]α∩α−1 α/α∩α−1 [a′]α∩α−1 ⇐⇒ aα a′ (since α∩α−1 is compatiblewith α) =⇒ h(a) ≤B h(a′) ⇐⇒ g([a]α∩α−1) ≤B g([a′]α∩α−1). �

Corollary 2.11 (Order Isomorphism Theorem). Let h:A → B be an order epimorphism ofρ-poalgebras, and let α = h−1(≤B) be the order kernel of h. Then A/α ∼= B; in particular,g:A/α ∼= B, where g([a]α∩α−1) = h(a) for all [a]α∩α−1 ∈ A/α ∩ α−1.

Proof. h is surjective, and by the theorem the mapping g: [a]α∩α−1 7→ h(a) is an orderhomomorphism from A/α to B such that h = n ◦ g. Moreover,

[a]α∩α−1 g−1(≤B) [a′]α∩α−1 ⇐⇒ g([a]α∩α−1) ≤B g([a′]α∩α−1)

⇐⇒ h(a) ≤B h(a′)

⇐⇒ a h−1(≤B) a′

⇐⇒ aα a′

⇐⇒ [a]α∩α−1 α/α ∩ α−1 [a′]α∩α−1 .

So the order kernel of g is α/α∩α−1. Hence g is an order isomorphism by Propostion 2.7. �January 11, 2004

8 DON PIGOZZI

Definition 2.12. A ρ-poalgebra A is a order subalgebra of ρ-poalgebra B, in symbolsA ⊆ B, if A ⊆ B, i.e., the underlying algebra of A is a subalgebra of the underlyingalgebra of B, and ≤A = ≤B ∩B2.

We say that A is generated by a set X of its elements if it is the smallest order subalgebraof B that includes X; it is easy to see that such a smallest subalgebra exists.

If h:A → B is an order homomorphism, the order subalgebra 〈h(A),≤B ∩h(A)2〉 of B iscalled the homomorphic image of h and is denoted by h(A). If h is an order monomorphism,then A is order isomorphic to h(A). In general, if A is order isomorphic to an ordersubalgebra of B we write A∼=;⊆B.

Definition 2.13. Let 〈Ai : i ∈ I 〉 be a system of ρ-poalgebras. By the order direct productof the system we mean the ρ-poalgebra∏

i∈I

Ai =⟨ ∏

i∈I

Ai,≤∏

Ai⟩,

where 〈 ai : i ∈ I 〉 ≤∏

Ai 〈 bi : i ∈ I 〉 if ai ≤Ai bi for all i ∈ I.

It is easy to check that the order direct product is a ρ-poalgebra, and that the projectionfunction πi:

∏j∈I Aj → Ai is an order epimorphism for each i ∈ I.∏

i∈I Ai together with the system 〈πi : i ∈ I 〉 is a product of Ai in the category ofρ-poalgebras. Thus given any system hi:B → Ai, i ∈ I, of order homomorphisms, there isa unique g:B →

∏i∈I Ai such that, for all i ∈ I, hi = πi ◦ g. g is denoted by

∏i∈I hi.

By a filter on a set I we mean a family F of subsets of I such that (1) I ∈ F , (2)J,K ∈ F implies J ∩K ∈ F , and (3) J ∈ F implies K ∈ F for every K ⊇ J . With eachfilter F on I is associated a ρ-qorder EF of the product

∏i∈I Ai by the condition that

〈 ai : i ∈ I 〉 EF 〈 bi : i ∈ I 〉 if { i ∈ I : ai ≤Ai bi } ∈ F . It is routine to check that EF is

indeed a ρ-qorder of∏

i∈I Ai. The quotient∏

i∈I Ai/EF is called an order reduced productof the system 〈Ai : i ∈ I 〉. A filter F over I is consistent if it does not contain the emptyset; it is an ultrafilter if is maximal and consistent. A reduced product by an ultrafilter iscalled an order ultraproduct.

There is one more construction that will be used in the sequel—the direct limit ; it isan order subalgebra of a special kind of reduced product. Let 〈Ai : i ∈ I 〉 be a systemof ρ-poalgebras indexed by a nonempty set I with an upward directed partial order ≤. Inaddition, let h = 〈hi,j : i ≤ j 〉 be a system of order epimorphisms indexed by the set ofpairs ≤ with the following properties: if i ≤ j, hi,j :Ai → Aj is an order epimorphismfrom Ai to Aj , hi,i is the identity map, and, if i ≤ j ≤ k, then hj,k ◦ hi,j = hi,k. Foreach i ∈ I, let [i) = { j ∈ I : i ≤ j }. Note that, since I is upward directed, for eachpair i, j there exists a k such that [k) ⊆ [i) ∩ [j). Let F be the set of all subsets of Ithat include a [i) for some i. Then F is a filter on I, and 〈 ai : i ∈ I 〉 EF 〈 bi : i ∈ I 〉iff, for some j, ai ≤Ai bi for all i ≥ j. Note that, again since I is upward directed,〈 ai : i ∈ I 〉EF ∩E−1

F 〈 bi : i ∈ I 〉 iff, for some j, ai = bi for all i ≥ j. Let B be the ordersubalgebra of

∏i∈I Ai consisting of all elements 〈 ai : i ∈ I 〉 with the property that, for

some i, hi,j(ai) = aj for all j ≥ i. B is clearly compatible with EF ∩E−1F in the sense

that if 〈 ai: i ∈ I 〉 ∈ B and 〈 ai: i ∈ I 〉EF ∩E−1F 〈 bi: i ∈ I 〉, then 〈 bi: i ∈ I 〉 ∈ B. So the

quotient B/EF of B by EF (more precisely its restriction to B2) is an order subalgebra ofJanuary 11, 2004


the order reduced product∏

i∈I Ai/EF . It is called the order direct limit of 〈Ai : i ∈ I 〉by h and is denoted by lim−→

hi∈IAi.

Let K be any class of ρ-poalgebras. The class of all order subalgebras of members of Kis denoted S(K), the class of all order homomorphic images of members of K is denotedby H(K), and the class of all ρ-poalgebras order isomorphic to an order direct productof systems of members of K is denoted by P(K). The classes of all ρ-poalgebras orderisomorphic respectively to an order reduced product, order ultraproduct, or order directlimit of members of K are denoted by Pr, Pu, and L−→(K). Finally, the class all orderisomorphic images of members of K is denoted by I(K).

Theorem 2.14. Let K be a class of ρ-poalgebras.(i) SH(K) ⊆ HS(K), PS(K) ⊆ SP(K), PH(K) ⊆ HP(K). P(K) ⊆ Pr(K), PPu(K) ⊆

PuP(K).(ii) K ⊆ L−→(K) ⊆ Pr(K).(iii) SPuP(K) = SPr(K).(iv) HSP and SPr(K) are closure operators on the class of all ρ-poalgebras.

Proof. The proofs of the inclusions of (i) are straightforward adaptations of the proofsof the corresponding results for unordered algebras. Consider for example the inclusionSH(K) ⊆ HS(K). Suppose h:C → B is an order homomorphism and A ⊆ B. Let D =〈h−1(A),≤C ∩h−1(A)2〉. Then D ⊆ C and h�D is an order epimorphism from D onto A.The two inclusions of (ii) are immediate consequences of the definition of direct limit. (iii)and (iv) follow easily from (i). �

2.1. Lattice of quasiorders and subdirect representation. Quasiorders play the rolein the theory of ρ-poalgebras that congruences play in the theory of algebras; like con-gruences, they form an algebraic lattice under set-theoretical inclusion. This view of qua-siorders was anticipated by [16, 17] in a more general context; see [15, page 32].

Theorem 2.15. Let A = 〈A,≤A〉 be a ρ-poalgebra. 〈A × A, Qordρ(A)〉 is an algebraicclosed set system.

Proof. Let K ⊆ Qordρ(A). Then⋂K is reflexive, transitive, and satisfies the ρ-tonicity

condition. Moreover, ≤A ⊆⋂K. If K is upward directed by inclusion, then

⋃K also has

these properties. �

It follows from this theorem that Qordρ(A) forms an algebraic lattice in which meet ofan arbitrary (possibly infinite) set of ρ-qorders is its set-theoretical intersection. The latticeof ρ-qorders of A, 〈Qordρ(A),

⋂,∨〉, is denoted by Qordρ.

Theorem 2.16. Let A be a ρ-poalgebra. If α, β ∈ Qordρ(A), then α ∨ β =⋃

n<ω(α ; β)n.More generally, for every K ⊆ Qordρ(A),∨

K =⋃{α0 ; α1 ; · · · ; αn−1 : n < ω, 〈α0, . . . , αn−1〉 ∈ Kn },

where α0 ; α1 ; · · · ; αn−1 is the relative product of the binary relations α0, . . . , αn−1, i.e.,the set of all pairs 〈a, b〉 such that there exist c0, . . . , cn such that

a = c0 α0 c1 α1 c2 . . . cn−1 αn−1 cn = b.

January 11, 2004

10 DON PIGOZZI

Proof. Let β =⋃{α0 ; α1 ; . . . ; αn−1 : n < ω, 〈α0, . . . , αn−1〉 ∈ Kn }. Clearly, β is reflexive

and transitive. We note that (α0 ; α1 ; · · · ; α0)−1 = α−1n−1 ; · · · ; α−1

1 ; α−10 . It is easy to see

that β−1 =⋃{α−1

0 ; α−11 ; . . . ; α−1

n−1 : n < ω, 〈α0, . . . , αn−1〉 ∈ Kn }.To see that β satisfies the ρ-tonicity condition, let σ be an operation, which for sim-

plicity we take of order 2,n whose first and second arguments are of negative and positivepolarity respectively. Assume a β−1 b and let α0, . . . , αn−1 ∈ K and c0, . . . , cn such thata = c0 α−1

0 c1 α−11 c2 · · · cn−1 α−1

n−1 cn = b Then for any d we have

σA(a, d) = σA(c0, d) α0 σA(c1, d) α1 σA(c2, d) · · ·σA(cn−1, d) αn−1 σA(cn, d) = σA(b, d).

Thus σA(a, d) β σA(b, d). This shows that σA is antimonotonic in the first argument withrespect to β. In a similar way it can be shown that it is monotonic is the second argument.Thus β satisfies the ρ-tonicity condition. So β is a ρ-qorder of A. Since it is clearly includedin every ρ-qorder that includes each α ∈ K, we get that β is the least upper bound of K inthe lattice Qordρ(A). �

Theorem 2.17 (Order Correspondence Theorem). Let A be an ρ-poalgebra and α ∈Qordρ(A). Then for every ρ-qorder β of A/α there is a unique β ∈ Qordρ(A) such that

β = β/α ={ ⟨

[a]α∩α−1 , [b]α∩α−1

⟩: 〈a, b〉 ∈ β

}.

The mapping β 7→ β is a lattice isomorphism between Qordρ(A/α) and the principal filterof Qordρ A generated by α.

Proof. Let β be a ρ-qorder of A/α = 〈A/α ∩ α−1, α/α ∩ α−1〉, i.e., a ρ-qordering ofA/α ∩ α−1 such that β ⊇ α/α ∩ α−1, and define β = { 〈a, b〉 : [a]α∩α−1 β [b]α∩α−1 }. Itis straightforward to verify that β is a ρ-qordering of A. For example, suppose σ is a bi-nary operation with negative ρ-polarity in the first argument, and let a β−1 b, and suppose[a]α∩α−1 β−1 [b]α∩α−1 . Then, for every c,

[σA(a, c)]α∩α−1 = σA/α∩α−1([a]α∩α−1 , [c]α∩α−1

)β σA/α∩α−1(

[b]α∩α−1 , [c]α∩α−1

)= [σA(b, c)]α∩α−1 .

So αA(a, c) β σA(b, c). It remains only to show that β ⊇ α. But this follows immediatelyfrom the inclusion β ⊇ α/α ∩ α−1.

Consider any γ ∈ Qordρ(A) that includes α. Then α ∩ α−1 is compatible with γ, andhence [a]α∩α−1 β [b]α∩α−1 iff a β b. It follows that γ/α ∩ α−1 =

{ ⟨[a]α∩α−1 , [b]α∩α−1

⟩:

〈a, b〉 ∈ γ}

is in Qordρ(A/α) and that, if γ, γ′ ∈ Qordρ(A) and γ/α ∩ α−1 = γ′/α ∩ α−1,then γ = γ′. So the mapping β 7→ β is a bijection between Qordρ(A/α) and the set of allβ ∈ Qordρ(A) such that β ⊇ α. It clearly preserves the lattice ordering. �

Remark 2.18. The key property of ρ-qorderings used in the above proof is that the sym-metrization of a ρ-qordering α is automatically compatible with every ρ-qordering thatincludes α. The deductive systems L considered in AAL with this property, that is theproperty that the Leibniz congruence of a L-filter F is compatible with every L-filter thatincludes F , are called protoalgebraic. The protoalgebraic deductive systems turn out tobe exactly those for which a correspondence theorem analogous to Theorem 2.17 holds,

January 11, 2004


and they seem to be the widest class of deductive systems for which a reasonable algebraictheory can be developed; see [6].

Definition 2.19. A ρ-poalgebra B is an order subdirect product of a system 〈Ai : i ∈ I 〉of ρ-poalgebras, in symbols B ⊆sd

∏i∈I Ai, if

(i) B ⊆∏

i∈I Ai, and(ii) πi:B → Ai is an order epimorphism for each i ∈ I.

Let K be a class of ρ-poalgebras. The class of all ρ-poalgebras isomorphic to a subdirectproduct of some system of members of K is denoted by Psd(K).

Proposition 2.20. B∼=;⊆sd∏

i∈I Ai iff there exists a 〈αi : i ∈ I 〉 ∈ Qordρ(B) such that

(i)⋂

i∈I αi = ≤B, and(ii) B/αi

∼= Ai for every i ∈ I.

Proof. =⇒ Let h:B ∼= C ⊆sd∏

i∈I Bi. For every i ∈ I, πi ◦ h: B → Ai is an orderepimorphism; let αi be its order kernel, i.e., αi = (πi ◦ h)−1(≤Ai). Then B/αi

∼= Ai by theorder isomorphism theorem.

b⋂

i∈Iαi b′ ⇐⇒(∀ i ∈ I

)(πi(h(b)) ≤Ai πi(h(b′))

)⇐⇒ h(b) ≤

∏Ai h(b′)

⇐⇒ h(b) ≤C h(b′)

⇐⇒ b ≤B b′.

So⋂

i∈I αi = ≤B.⇐= By the categorical product property h =

∏i∈I ni, where ni:Ai → Ai/αi is the

natural order epimorphism, is an order homomorphism h:B →∏

i∈I B/αi; note that h(b) =〈 [b]αi∩α−1

i: i ∈ I 〉.

b h−1(≤∏

Ai/αi) b′ ⇐⇒ 〈 [b]αi∩α−1i

: i ∈ I 〉 ≤∏

B/αi 〈 [b′]αi∩α−1i

: i ∈ I 〉

⇐⇒(∀ i ∈ I

)([b]αi∩α−1

iαi/αi ∩ α−1

i [b′]αi∩α−1i

)⇐⇒ (∀ i ∈ I)(b αi b

′)

⇐⇒ b⋂i∈I

αi b′

⇐⇒ b ≤B b′.

So the order kernel of h is ≤B. Hence B ∼= h(B) ⊆∏

i∈I B/αi. Since πi ◦ h:B → B/αi isan order epimorphism for each i ∈ I we get that h(B) ⊆sd

∏i∈I B/αi. �

Definition 2.21. A ρ-poalgebra B is order subdirectly irreducible if B∼=;⊆sd∏

i∈I Ai im-plies that B ∼= Ai for some i ∈ I.

Theorem 2.22 (Order Subdirect Representation Theorem). Every ρ-poalgebra is isomor-phic to an order subdirect product of order subdirectly irreducible ρ-poalgebras.

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12 DON PIGOZZI

Proof. Let A be a ρ-poalgebra. For each pair a, b of elements of A such that a �A b, let αa,b

be a maximal ρ-qorder of A such that 〈a, b〉 /∈ αa,b; such a maximal ρ-qorder exists by Zorn’slemma. αa,b is completely meet irreducible in the lattice Qordρ(A), for suppose 〈βi : i ∈ I 〉is any system of ρ-qorders such that αa,b ⊂ βi for all i ∈ I. Then a βi b. Thus αa,b 6=

⋂i∈I βi.

Clearly ≤A =⋂{αa,b : a �A b }. So by the Prop 2.20 A∼=;⊆sd

∏a�Ab A/αa,b. Suppose

A/αa,b∼=;⊆sd

∏i∈I Bi. By Prop. 2.20 and the order correspondence theorem there is a

system 〈βi : i ∈ I 〉 of ρ-qorders of A such that αa,b =⋂

i∈I βI and Ai/βi∼= Bi. Since αa,b

is completely join-irreducible, αa,b = βi for some i. So A/αa,b∼= Bi, and hence each A/αa,b

is order subdirectly irreducible. �

3. Ordered Equational Logic

The set of terms over the signature Σ in the variables X is denoted by TeΣ(X). Either of“Σ” or “X” may be omitted if they are clear from context or irrelevant. By an inequationwe mean an ordered pair of terms 〈t, s〉, which we normally write in the form t 4 s. A quasi-inequation is a non-empty sequence of inequations, written t0 4 s0, . . . , tn−1 4 sn−1 → u 4v. The inequations t0 4 s0, . . . , tn−1 4 sn−1 are called the premisses and u 4 v theconclusion of the quasi-inequation. Inequations are identified with quasi-inequations withan empty set of premisses.

We write a term t in the form t(x0, . . . , xn−1) to indicate that the variables occurring int all appear in the list x0, . . . , xn−1; we often use the abbreviation x for x0, . . . , xn−1. Lett(x) 4 s(x) be an inequation and A a ρ-poalgebra. An assignment a ∈ An of elements ofA to the variables x is said to satisfy t(x) 4 s(x) in A if tA(a) ≤A sA(a). It satisfies aquasi-inequation if it either fails to satisfy at least one of the premisses, or it satisfies theconclusion.

A quasi-inequation is said to be a quasi-inidentity of a ρ-poalgebra A if it is satisfiedby every assignment of elements of A to the variables; in this case we also say that A is amodel of the quasi-inequation. As a special case we have the definition of the inidentitiesof A. For any set Q of quasi-inequations or inequations, the class of models of all membersof Q is denoted by Mod(Q).

Definition 3.1. A class Q of ρ-poalgebras is called a ρ-ordered quasivariety, a ρ-quasi-povariety for short, if V = Mod(Q) for some set of quasi-inequations Q. The class of modelsof a set of inequations is called a ρ-ordered variety, a ρ-povariety. If ρ is the completelypositive polarity, Q is called simply a quasi-povariety or povariety.

Example 3.2 (Lattices As an Povariety). The signature is Σ = {∧,∨} and, as the term“povariety” implies, the polarity is completely positive. The povariety of lattices is definedby the following six extralogical inidentities.

x ∧ y 4 x, x ∧ y 4 y,(1)

x 4 x ∨ y, y 4 x ∨ y,(2)

x 4 x ∧ x, x ∨ x 4 x.(3)

We recall the logical inidentities and quasi-inidentities that are automatically included inthe definition of any povariety.January 11, 2004


refl x 4 x,tran x 4 y, y 4 z =⇒ x 4 z,

toni∧ x1 4 x2, y1 4 y2 =⇒ x1 ∧ y1 4 x2 ∧ y2;toni∨ x1 4 x2, y1 4 y2 =⇒ x1 ∨ y1 4 x2 ∨ y2.It is obvious that if 〈A,∧,∨〉 is a lattice in the usual sense, then

⟨〈A,∧,∨〉, ≤

⟩is a

model of the inidentities (1)–(3). Conversely, let⟨〈A,∧,∨〉, ≤

⟩be a poalgebra satisfying

the inidentities (1)–(3), and let a, b ∈ A. a ∧ b ≤ a, b by (1). Suppose c ≤ a, b. Then by (2)and (mono∧) we get c ≤ c∧ c ≤ a∧ b. So a∧ b is the greatest lower bound of a, b. Similarly,a ∨ b is the least upper bound of a, b. So 〈A,∧,∨〉 is a lattice in the ordinary sense.

Example 3.3 (Partially Ordered Left-Residuated Monoids (POLRMS)). Recall that theseare ρ-poalgebras

⟨〈A, ·,→, 1〉,≤

⟩where ρ(·, 0) = ρ(·, 1) = ρ(→, 1) = + and ρ(→, 0) = −,

and such that 〈A, ·, 1〉 is a monoid and the residuation condition holds: for all a, b, z ∈ A,z · a ≤ b iff z ≤ a → b.

Proposition 3.4. POLRM is a ρ-povariety defined by the following set of (extra-logical)inidentities.

(x · y) · z 4< x · (y · z)(4)

1 · x 4< x(5)

x · 1 4< x(6)

x · (x → y) 4 y(7)

y 4 x → x · y(8)

Proof. That each POLRM is a ρ-algebra satisfying the inidentities (4)–(6) is clear. (7) and(8) follow respectively from the inidentities x → y 4 x → y and x · y 4 x · y by residuation.

Suppose⟨〈A, ·,→, 1〉,≤

⟩is a ρ-poalgebra satisfying (4)–(8). 〈A, ·, 1〉 is a monoid by (4)–

(6). We verify the residuation condition: Suppose a ≤ b → c. Then b · a ≤ b · (b → c) ≤ c.Suppose b · a ≤ c. Then a ≤ b → b · a ≤ b → c. �

Partially ordered groups can be viewed as a special kind of POLRM. A partially orderedgroup is a group 〈G, ·,−1, e〉 with a partial ordering ≤ of its universe with respect to which· is monotone in both arguments. Define x → y as x−1 · y. Then

⟨〈G, ·,→, e〉,≤

⟩is a

POLRM, and the class of all partially ordered groups in this sense forms a sub-ρ-povarietyof POLRM.

Remark 3.5. It should be noted that there is no equality symbol in the language oflattices as ordered algebras; this is an important feature of the metamathematics of ρ-poalgebras in our treatment. Inequational logics from this point of view can be treatedas universal Horn theories with a single, binary predicate and without equality. Theseare in effect the 2-dimensional deductive systems of AAL. When viewed as universal Horntheories in this way, the most general models of the inequational logic of lattices are of theform

⟨〈A,∧,∨〉,≤

⟩, where ≤ is a not necessarily antisymmetric quasi-ordering of 〈A,∧,∨〉

satisfying the inidentities (1)–(2), i.e., its symmetrization 〈A,∧,∨〉/ ≤ ∩ ≥ is a lattice inthe ordinary sense.

The most general models of an arbitrary inequational logic are algebras with a distin-guished ρ-qordering, but we restrict our attention to those special models for which theJanuary 11, 2004

14 DON PIGOZZI

quasi-ordering is antisymmetric. These are the so-called reduced models of the logic whenconsidered a 2-dimensional deductive system. Recall that the symmetrization ≤ ∩ ≥ of agiven ρ-qordering ≤ is the largest congruence compatible with ≤. More generally, for eachmodel of an arbitrary 2 dimensional deductive system, more generally still, for a k-deductivesystem, there is a largest congruence that is compatible with the so-called designated filter ofthe model; this is a binary or k-ary relation on the universe of the algebra that correspondsto the ρ-qorder of the nonreduced models of inequation logic. This congruence is called theLeibniz congruence of the model, and the model is reduced if its Leibniz congruence is theidentity.

Notice that the logical axioms of inequational logic except the first are quasi-inequations.Thus strictly speaking, considered as a universal Horn theory without equality, the reducedmodels of a set of inequations form a quasivariety. But following the example of equationallogic we refer to the reduced models of an inequational logic in which there are no properquasi-inequations among the extralogical axioms as a variety.

Example 3.6 (The Variety of (Unordered) Algebras as a Quasi-povariety). It is instructiveto observe that ordinary equational logic can be viewed in a natural way as the inequationallogic with a single extralogical axiom, namely the symmetry law x 4 y → y 4 x. Itsnonreduced models are of the form 〈A, α〉 where α is a congruence relation on A, and itsreduced models are 〈A,∆A〉, which can be indentified with its underlying algebra A.

By simply making the extralogical axiom of antisymmetry a logical axiom we get Birkhoff’swell known equational logic.

Remark 3.7. The above two examples and the intervening remark illustrate two significantfeatures of our approach to the metamathematics of ordered algebras. One is the isolationof equality from inequality in the formal development. Equality enters only peripherallyby means of the Leibniz congruence and the restriction to reduced models. In the case ofinequational logic the Leibniz equality is definable by the two inequations x 4 y, y 4 x, butin an arbitrary k-deductive system it may not be so readily definable. Those systems forwhich it is are called equivalential logics and constitute an extensively investgated propersubclass of the class of protoalgebraic logics.

The other point the last example illustrates is that, in our approach, the theory of orderedalgebras is developed as a logical system apart from the equational logic of Birkhoff ratherthan by building on it—in short the two theories are developed in parallel rather than inseries. A lot of work on theory of ordered algebras can be found in the literature basedon the alternative approach, and most of the results we obtain here are not significantlydifferent from what can be found in the literature, and no claim is made for their novelty.However we believe our approach is smoother and more natural exactly because of the factit parallels traditional equational logic rather than builds on it. Furthermore, it raises newquestions that lead to new and we thing interesting results of a different character. Someof will be seen in the next two sections.

Definition 3.8. Let K be an arbitrary class of ρ-poalgebras. For each ρ-poalgebra A (notnecessarily in K) we define

QordKρ (A) = {α ∈ Qordρ(A) : A/α ∈ K }.

January 11, 2004


The elements of QordKρ (A) are called the K-ρ-qorders of A.

Proposition 3.9. Let K be a class of ρ-poalgebras, and let A be a ρ-algebra, not necessarilyin K.

(i) Assume Psd(K) ⊆ K. Then QordKρ (A) is closed under arbitrary intersection. It

follows that QordKρ (A) contains a smallest ρ-qorder.

(ii) Assume H(K) ⊆ K. If α ∈ QordKρ (A), then β ∈ QordK

ρ (A) for every β ∈ Qordρ(A)such that β ⊇ α.

(iii) Assume both Psd(K) ⊆ K and H(K) ⊆ K. Then QordKρ (A) is a principal filter of

Qordρ(A).

Proof. (i) Let {αi : i ∈ I } ⊆ QordKρ (A). By Proposition 2.20 and the order correspon-

dence theorem, A/⋂

i∈I αi∼=;⊆sd

∏{A/αi : i ∈ I } ∈ Psd{A/αi : i ∈ I } ⊆ PsdK ⊆ K.

(ii) By the order homomorphism theorem, A/β is a homomorphic image of A/α. Thus,since A/α ∈ K by the assumption, and K is closed under homomorphic images by hypoth-esis, we have A/β ∈ K and hence β ∈ QordK

ρ (A).(iii) is an immediate consequence of (i) and (ii). �

Definition 3.10. Let K be a ρ-quasivariety, and let A be an arbitrary ρ-poalgebra, notnecessarily in K. Let R ⊂ A2. By the K-ρ-qorder of A generated by R, in symbols ΦK

ρ (R),we mean the smallest K-ρ-qorder of A that includes R, i.e., ΦK

ρ (R) =⋂{α ∈ QordK

ρ (A) :R ⊆ α }.

ΦKρ (R) always exists if Psd(K) ⊆ K.

Definition 3.11. Let K be a class of ρ-poalgebras. A ρ-poalgebra is said to be freelygenerated over K by a set X if X generates A and, for every B ∈ K and every mappingh:X → B, there is an order homomorphism from A into B that extends h.

The extension is unique because of the assumption that X generates A; it is normallyrepresented by the same symbol as the original map.

For any set of variables X the algebra of terms over X, or term algebra over X, is thealgebra Te(X) whose universe is Te(X) and such that, for each σ ∈ Σn, the operationσTe(X) takes the term σ(t0, . . . , tn−1) as value for each choice of arguments t0, . . . , tn−1, i.e.,σTe(X)(t0, . . . , tn−1) = σ(t0, . . . , tn−1). Te(X) is freely generated by X over the class ofall Σ-algebras. We note that the value tA(a0, . . . , an−1) a term t(x0, . . . , xn−1) takes in analgebra A under the assignment of a0, . . . , an−1 to the variables coincides with the imageh(t(x0, . . . , xn−1)) of t(x0, . . . , xn−1) under any homomorphism h:Te(X) → A such thath(x0) = a0, . . . , h(xn−1) = an−1, i.e., h(t(x0, . . . , xn−1) = tA(h(x0), h(x1), . . . , h(xn−1)).

The ρ-poalgebra 〈Te(X),∆Te(X)〉 over the term algebra is denoted by Te(X). From theabove remarks it follows that, for any set X of cardinality κ, Te(X) is freely generatedover the class of all ρ-poalgebras by the set X.

Theorem 3.12. Let K be a class of ρ-poalgebras such that SP(K) ⊆ K. Then for eachcardinal κ there exists a ρ-poalgebra in K that is freely generated over K by a set of cardinalityκ. More specifically, let X be any set of cardinality κ, and let α be the smallest memberof QordK

ρ (Te(X)). Then Te(X)/α is a member of K and is freely generated over K byX/α ∩ α−1 = { [x]α∩α−1 : x ∈ X }.January 11, 2004

16 DON PIGOZZI

Any two ρ-polagebras in K that are freely generated over K by sets of the same cardinalityare isomorphic.

Proof. Te(X)/α ∈ K by definition of α. Let A ∈ K and h:X/α ∩ α−1 → A. Let h′ bethe unique order homomorphism from Te(X) into A such that h′(x) = h([x]α∩α−1) foreach x ∈ X. ordker(h′) ∈ QordK

ρ (F), so α ⊆ ordker(h′). Thus by the order homomorphismtheorem there is a unique order homomorphism g:Te(X)/α → A such that, for each x ∈ X,g([x]α∩α−1) = h(x).

Suppose A and B are freely generated over K by sets of the same cardinality. Then anybijection between the sets of free generators extends to an order isomorphism between Aand B. �

Lemma 3.13. Let K be a class of ρ-poalgebras such that SP(K) ⊆ K. Let α be the smallestmember of QordK

ρ (Te(X)). Then, for any pair of terms t, s over the variables X, t 4 s isan inidentity of K iff t αs.

Proof. =⇒ Assume t(x0, . . . , xn−1) 4 s(x0, . . . , xn−1) is an inidentity of K. SinceTe(X)/α ∈ K, we have

[t(x0, . . . , xn−1)]α∩α−1 = tTe(X)/α∩α−1([x0]α∩α−1 , . . . , [xn−1]α∩α−1

)≤Te(X)/α sTe(X)/α∩α−1(

[x0]α∩α−1 , . . . , [xn−1]α∩α−1

)= [s(x0, . . . , xn−1)]α∩α−1 .

But ≤Te(X)/α = α/α ∩ α−1. So t(x0, . . . , xn−1) α (x0, . . . , xn−1).⇐= Assume t α s. Let A be an arbitrary member of K and h:Te(X) → A any order

homomorphism. α ⊆ ordker(h) since ordker(h) ∈ QordTe(X)ρ . So t h−1(≤A) s, and hence

h(t) ≤A h(s). Since this holds for any A ∈ K and h:Te(X) → A, t 4 s is an inidentity ofK. �

Theorem 3.14 (Order H-S-P Theorem). A class K of ρ-poalgebras is an order varietyiff it is closed under the formation of order homomorphic images, subalgebras, and directproducts, i.e., iff HSP(K) = K.

Proof. =⇒ Assume I is a set of inidentities such that K = Mod(I). Consider anyA ∈ H(K) and order homomorphism h:Te(X) → A. There is a B ∈ K and an or-der epimorphism g:B → A. Since g is surjective, there exists an order homomorphismf :Te(X) → B such that h = g ◦ f . Let t 4 s be an inidentity of I. Since B ∈ K we havef(t) ≤B f(s), and since g is an order homomorphism, g(f(t) ≤A g(f(s)), i.e., h(t) ≤A h(s).This shows that H(K) ⊆ K. The proofs S(K) ⊆ K and P(K) ⊆ K are obtained in a similarmanner; we leave the details to the interested reader. We only mention that the proof inthe direct product case uses the fact the the direct product is a product in the category ofρ-poalgebras.⇐= Assume HSP(K) = K, so in particular H(K) ⊆ K and SP(K) = K. Let I be

the set of all inidentitites of K. We will show that K = Mod(I), and since the inclusionfrom left to right is obvious, it suffices to show that Mod(I) ⊆ K. Suppose A ∈ Mod(I).Choose X large enough so that an order epimorphism h:Te(X) → A exists. Let α bethe smallest member of QordK

ρ (Te(X)). Then for all terms t, s we get, using Lemma 3.13,January 11, 2004


that t α s =⇒ t 4 s ∈ I =⇒ h(t) ≤A h(s). Hence α ⊆ ordker(h), and so by the orderhomomorphism theorem there is a g:Te(X)/α → A such that h = g ◦ n. g is an orderepimorphism since h is, and so A is a homomorphic image of Te(X)/α. But the latterρ-poalgebra is in K by definition of α. Thus A ∈ H(K) = K. �

In the last part of the section we obtain an analogous operator-theoretic characterizationfor order quasivarieties that parallels Mal’cev’s well known characterization of quasivarietiesof unordered quasivarieties as refined in [17, 18].

Lemma 3.15. Let K be a class of ρ-poalgebras such that SP(K) ⊆ K. Let t0 4 s0, . . . , tn−1 4sn−1 be a finite set of inequations over the variables X. ΦK

ρ ({ ti 4 si : i < n }) is the smallestmember α of QordK

ρ

(Te(X)

)such that ti α si for all i < n. For any inequality u 4 v, the

quasi-inequation

(9) t0 4 s0, . . . , tn−1 4 sn−1 → u 4 v

is a quasi-inidentity of K iff u ΦKρ ({ ti 4 si : i < n }) v.

Proof. Let α = ΦKρ ({ ti 4 si : i < n }), and let x = x0, . . . , xm−1 be the variables occurring

in (9). Let [x]α∩α−1 denote the sequence [x0]α∩α−1 , . . . , [xn−1]α∩α−1 of elements of Te(X)/α.We note first that, by definition of α, [ti(x)]α∩α−1 ≤Te(X)/α [si(x)]α∩α−1 , and hence

(10) tTe(X)/α∩α−1

i

([x]α∩α−1

)≤Te(X)/α s

Te(X)/α∩α−1

i

([x]α∩α−1

), for each i < n.

=⇒ Assume (9) is a quasi-inidentity of K. Then by (10) and the fact that Te(X)/α ∈ K

we have that uTe(X)/α∩α−1([x]α∩α−1) ≤Te(X)/α vTe(X)/α∩α−1

([x]α∩α−1), and hence u(x) α v(x).⇐⇒ Assume u(x) α v(x). Let A ∈ K and let h:Te(X) → A be any order ho-

momorphism such that tAi(h(x)

)≤A sA

i

(h(x)

)for each i < n. Then α ⊆ h−1(≤A)

by definition of α since h−1(≤A) ∈ QordKρ (Te(X)). So u(x) h−1(≤A) v(x) and hence

uA(h(x)) ≤A vA(h(x)). Thus (9) is a quasi-inidentity of K. �

Lemma 3.16. Let A be a ρ-poalgebra and K a set of ρ-qorders of A that is upward directedby inclusion so that

⋃K is also a ρ-qorder of A. Then A/

⋃K is isomorphic to the direct

limit of the system of ρ-poalgebras 〈A/α : α ∈ K} by the system of order epimorphisms

h = 〈hα,β : A/α → A/β : α, β ∈ K, α ⊆ β 〉,where hα,β([a]α∩α−1 = [a]β∩β−1 for all [a]α∩α−1 ∈ A/α ∩ α−1.

Proof. Let F = { G ⊆ K : (∃α)((α] ⊆ G

)}, and let g:A → lim−→

hα∈KA/α ∩ α−1 be the map

in such that, for each a ∈ A, g(a) =[⟨

[a]α∩α−1 : α ∈ K⟩]

EF∩E−1F

. It is clear that g maps

into lim−→hα∈KA/α ∩ α−1. To see it is an order homomorphism, consider any σ ∈ Σ, of rank

January 11, 2004

18 DON PIGOZZI

n say, and any a0, . . . , an−1 ∈ A.

g(σA(a0, . . . , an−1)

)=

[〈 [σA(a0, . . . , an−1)]α∩α−1 : α ∈ K 〉

]EF∩E−1

F

=[⟨

σA/α∩α−1([a0]α∩α−1 , · · · , [an−1]α∩α−1) : α ∈ K

⟩]EF∩E−1

F

= σlim−→

hα∈KAα

([⟨[a0]α∩α−1 :α ∈ K

⟩]EF∩E−1

F, . . .

. . . ,[⟨

[an−1]α∩α−1 : α ∈ K⟩]

EF∩E−1F

)= σ

lim−→hα∈KAα

(g(a0), . . . , g(an−1)

).

And, for all α, β ∈ K,

a ≤A b =⇒(∀α ∈ K

)([a]α∩α−1 ≤A/α [b]α∩α−1

)⇐⇒

⟨[a]α∩α−1 : α ∈ K

⟩≤

∏A/α

⟨[b]α∩α−1 : α ∈ K

⟩=⇒

[⟨[a]α∩α−1 : α ∈ K

⟩]EF∩E−1

F≤lim−→

hα∈KAα

[⟨[b]α∩α−1 : α ∈ K

⟩]EF∩E−1

F

⇐⇒ g(a) ≤lim−→hα∈KAα g(b).

So g is an order homomorphism. To see it is surjective, recall first of all that every elementof the direct limit is of the form

[⟨[aα]α∩α−1 : α ∈ K

⟩]EF∩E−1

F, wheren, for some γ ∈ K,

hγ,α([aγ ]γ∩γ−1) = [aα]α∩α−1 for every α ∈ K such that γ ⊆ α. Let b = aγ . For, for every α ⊇γ, we have [aα]α∩α−1 = hγ,α[b] = [b]α∩α−1 . So

⟨[b]α∩α−1 : α ∈ K

⟩EF ∩E−1

F⟨[aα]α∩α−1 :

α ∈ K⟩, and hence, g(b) =

[⟨[aα]α∩α−1 : α ∈ K

⟩]EF∩E−1

F. So g is an order epimorphism.

Finally, we show that its order kernel is⋂K. Let a, b ∈ A.

g(a) ≤lim−→hα∈KAα g(b) ⇐⇒

[⟨[a]α∩α−1 : α ∈ K

⟩]E∩E−1 ≤

lim−→hα∈KAα

[⟨[b]α∩α−1 : α ∈ K

⟩]E∩E−1

⇐⇒(∃ γ

)(∀α ⊇ γ

)([a]α∩α−1 ≤A/α [b]α∩α−1

)⇐⇒

(∃ γ

)(∀α ⊇ γ

)(aα b

)⇐⇒ a

⋃K b.

So, the order kernel of g is⋃K, and hence A/

⋃K ∼= lim−→

hα∈KAα by the order isomorphism

theorem. �

Theorem 3.17 (Order S-L-P Theorem). A class K of ρ-poalgebras is an ρ-quasi-povarietyiff it is closed under the formation of subalgebras, direct limits, and products i.e., iffS L−→P(K) = K.

Proof. =⇒ The proof that a ρ-quasi-povariety is a closed under order subalgebras and re-duced products is straightforward. Its closed under order direct limits and product becausethey are special kinds of reduced products.⇐= Assume S L−→P(K) = K. Let Q be the set of all quasi-inidentities K. We will show

that K = Mod(Q). The inclusion from left to right is obvious. Suppose A ∈ Mod(Q). LetX be a large enough set of variables such that there is a epimorphism h:Te(X) → A. Letα be the order kernel of h.

January 11, 2004


Recall that, for each R ⊆ω α (i.e, for each finite subset of α), ΦKρ (R), the ρ-K-qorder

generated by R, is the smallest K-ρ-qorder of Te(X) that includes R. (It exists by Propo-sition 3.9(i) since SPsd(K) ⊆ K by assumption.) Let R = { t0 4 s0, . . . , tn−1 4 sn−1},and let u ΦK

ρ (R) v. Then, by Lemma 3.15, t0 4 s0, . . . , tn−1 4 sn−1 → u 4 v is a quasi-inidentity of K. By hypothesis it is also a quasi-inidentity of A. h(ti) = h(si) for alli < n since R is included in the order kernel α of h. Hence h(u) = h(v), i.e., u α v. Thisshows that ΦK

ρ (R) ⊆ α for every R ⊆ω α. Thus α =⋃

R⊆ωα ΦKρ (R). Clearly the set of

K-ρ-qorders ΦKρ (R) are upward directed by inclusion. Thus by Lemma 3.16 A is the direct

limit of the system⟨Te(X)/ΦK

ρ (R) : R ⊆ω α⟩. Since each Te(X)/ΦK

ρ (R) ∈ K, we getA ∈ L−→(K) = K. �

It follows immediately that K is a ρ-quasi-povariety iff SPr(K) = K iff SPuP(K) = K.

4. Algebraizable ρ-Povarieties

We define for an arbitrary ρ-poalgebra A = 〈A,≤A〉 the algebra reduct of A, in symbolsAlg(A), to be the underlying algebra with the ρ-pordering replaced with the identityrelation, i.e., Alg(A) = 〈A,∆A〉, which for convenience we identify with A. (In contrast toconventional model theory the equality predicate is not assumed here to be a basic part ofthe definition of an algebra.) For any class of ρ-poalgebras, Alg(K) = {Alg(A) : A ∈ K }.

It turns out that Alg(K) is a quasivariety for every ρ-quasi-povariety K. This can beobtained directly for the well-known result of Mal’cev that the class of all substructures ofa reduct of a universal Horn class is again a universal Horn class; it is easy to see that Alg(K)is closed under the formation of subalgebras. Alternatively, using another, related resultof Mal’cev, the result can be verified by observing that Alg(K) is closed under subalgebrasand reduced products.

Let Q be a quasivariety of algebras, and let A be an algebra not necessarily a memberof Q. Those congruences α of A such that A/α ∈ Q are called Q-congruences. We takeCoQ(A) = {α ∈ Co(A) : A/α ∈ Q }.

Proposition 4.1. Assume K is a ρ-quasi-povariety. Let A be a Σ-algebra (not neces-sarily in K). Then mapping α 7→ α ∩ α−1 is a surjective map from QordK

ρ (〈A,∆A〉) ontoCoAlg(K)(A).

Proof. If α ∈ QordKρ (〈A,∆A〉), then A/α = 〈A/α ∩ α−1, α/α ∩ α−1〉 ∈ K and hence

α ∩ α−1 ∈ CoAlg(K). It remains to show the mapping is surjective.Suppose ϑ ∈ CoAlg(K)(A). A/ϑ ∈ Alg(K) by hypothesis, so there exists a ρ-ordering ≤

of A/ϑ. Let α = n−1(≤), where n:A → A/ϑ is the natural map. For all a, b ∈ A wehave a α ∩ α−1 b iff a n−1(≤) ∩ n−1(≥) b iff a n−1(≤≥) b iff a n−1(∆A/ϑ) b iff aϑ b.So α ∩ α−1 = ϑ, and hence α/α ∩ α−1 = ≤. So 〈A,∆A〉/α = 〈A/α ∩ α−1, α/α ∩ α−1〉 =〈A/ϑ,≤〉 ∈ K. We conclude that α ∈ QordK

ρ (〈A,∆A〉), and because α ∩ α−1 = ϑ, we seethat the map α 7→ α ∩ α−1 does indeed map Qordρ(〈A,∆A) surjectively onto CoAlg(K)(A).

�

If K is algebraizable, then an axiomatization of Alg(K) can be obtained from the set ofdefining equations, as shown in Proposition 4.6 below.

January 11, 2004

20 DON PIGOZZI

Definition 4.2. Let A = 〈A,≤A〉 be a ρ-poalgebra and let (T (x, y) ≈ S(x, y)) = { ti(x, y) ≈si(x, y) : i < n } be a finite set of equations in two variables. ≤A is definable by T ≈ S if,for all a, b ∈ A,

a ≤A b ⇐⇒ TA(a, b) = SA(a, b),where there expression “TA(a, b) = SA(a, b)” is shorthand for “

(∀ i < n

)(tAi (a, b) =

sAi (a, b)

)”.

A ρ-quasi-povariety K is algebraizable if there exists a finite set of formulas T ≈ S in twovariables such that, for every A in K, T ≈ S defines ≤A. T ≈ S is called a defining set ofequations for K.

Proposition 4.3. Assume K is an algebraizable ρ-quasi-povariety with defining equationsT ≈ S. Then, for every A ∈ K and every α ∈ QordK

ρ (A), we have for all a, b ∈ A:

aα b ⇐⇒ TA(a, b) α ∩ α−1 SA(a, b),

where the expression “TA(a, b) α ∩ α−1 SA(a, b)” is shorthand for the expression“(∀ t ≈ s ∈ (T ≈ S)

)(tA(a, b) α ∩ α−1 sA(a, b)

)”.

Proof.

aα b ⇐⇒ [a]α∩α−1 ≤A/α∩α−1[b]α∩α−1

⇐⇒ TA/α∩α−1([a]α∩α−1 , [b]α∩α−1

)= SA/α∩α−1(

[a]α∩α−1 , [b]α∩α−1

)⇐⇒ TA(a, b) α ∩ α−1 SA(a, b).

�

In every ρ-poalgebra equality is definable in terms of the order by symmetrization:

a = b ⇐⇒ a ≤A b and b ≤A a.

(In the terminology of AAL this means that the inequational logic of each ρ-quasi-povarietyis protoalgebraic, a fact previously noted in Remark 3.7.) If ≤A is definable by T ≈ S,then

(11) a = b ⇐⇒ TA(a, b) = SA(a, b) and TA(b, a) = SA(b, a).

From (11) it follows that, if K is algebraizable with defining equations T ≈ S, then theequations in T (x, x) ≈ S(x, x) are identities of Alg(K) and the quasi-equation

(T (x, y) ≈ S(x, y)), (T (y, x) ≈ S(y, x)) → x ≈ y

is a quasi-identity of Alg(K).

Corollary 4.4. Assume K is an algebraizable ρ-quasi-povariety.(i) For every A ∈ Alg(K) there exists a unique ρ-pordering ≤ of A such that 〈A,≤〉 ∈ K.(ii) For every A ∈ K, the mapping α 7→ α ∩ α−1 from QordK

ρ (A) to Co(A) is injective.�

Example 4.5. Each of {x ∧ y ≈ x} and {x ∨ y ≈ y} is a defining set of equations for thepovariety of lattices.

The ρ-povariety of POLRMs is not algebraizable. In fact, the ρ-subpovariety of partiallyordered groups is not algebraizable. To show this we need only exhibit a group with twoJanuary 11, 2004


distinct partial orderings, for example⟨〈Z,+,−, 0〉,≤

⟩and

⟨〈Z,+,−, 0〉,∆Z

⟩, where ≤ is

the natural ordering of the integers.

Proposition 4.6. Assume K is an algebraizable ρ-quasi-povariety. If T ≈ S is a set ofdefining equations for K, then the quasivariety Alg(K) is defined by the following identitiesand quasi-identities, where I and Q are respectively any any set of inidentities and quasi-inidentities that together define K.

T (x, x) ≈ S(x, x).(12)

T (x, y) ≈ S(x, y), T (y, z) ≈ S(y, z) → T (x, z) ≈ S(x, z).(13)

This expression represents the set of m quasi-equations all of which have the same an-tecedent, the conjunction of the 2m equations in T (x, y) ≈ S(x, y) and T (y, z) ≈ S(y, z),and one of the equations in T (x, z) ≈ S(x, z) as consequent (m is the number of equationsin T ≈ S).

T (x, y) ≈ S(x, y) → T(σ(z<i, x, z>i

)≈ S

(σ(z<i, y, z>i

)for each σ ∈ Σn and i < n such that ρ(σ, i) = +,

(14)

T (y, x) ≈ S(y, x) → T(σ(z<i, x, z>i

)≈ S

(σ(z<i, y, z>i

)for each σ ∈ Σn and i < n such that ρ(σ, i) = −,

(15)

T (x, y) ≈ S(x, y), T (y, x) ≈ S(x, y) → x ≈ y.(16)

T (t, s) ≈ S(t, s) for every t 4 s in I.(17)

T (t0, s0) ≈ S(t0, s0), . . . ,T (tn−1, sn−1) ≈ S(tn−1, sn−1) → T (u, v) ≈ S(u, v)for every t0 4 s0, . . . , tn−1 4 sn−1 → u 4 v in Q

(18)

Proof. Let E be the set of equations and quasi-equations (12)–(18). Clearly Alg(K) ⊆Mod(E) (see the remark following Proposition 4.3). Suppose A ∈ Mod(E). Define ≤ ⊆ A2

by the condition that a ≤ b iff TA(a, b) = SA(a, b). ≤ is a ρ-qordering of A by (12)–(15),a partial ordering by (16), and 〈A,≤〉 ∈ Mod(K) by (17) and (18). So A ∈ Alg(K). �

Taking K to be the povariety of lattices, we see that (16) takes the form

x ∧ y ≈ x, y ∧ x ≈ y → x ≈ y.

This is a consequence of the lattice identity x ∧ y ≈ y ∧ x. The quasi-identities (13)–(15)can also be replaced by identities. So Alg(K) is a variety, in fact the variety of lattices.

Proposition 4.7. Let K be an algebraizable ρ-quasi-povariety. Then, for each A ∈ K, themap α 7→ α ∩ α−1 is an isomorphism between the lattices QordK

ρ (A) and CoAlg(K)(A).

Proof. �

Theorem 4.8. Let K be a ρ-quasi-povariety. The following are equivalent.(i) K is algebraizable.(ii) For each A ∈ K, α 7→ α ∩ α−1 is an injective map from QordK

ρ (A) to CoAlg(K)(A).(iii) For each A ∈ K, α 7→ α ∩ α−1 is an isomorphism between QordK

ρ (A) and CoAlg(K)(A).

January 11, 2004

22 DON PIGOZZI

Proof. (i) ⇒ (ii): by Corollary 4.4.(ii) ⇒ (iii) Assume the map α 7→ α ∩ α−1 is injective. It is clearly order preserving.

Since ≤A is the unique ρ-pordering ≤ of A such that 〈A,≤〉 ∈ K, we have that QordKρ (A) =

QordKρ (〈A,∆A〉). Hence by Proposition 4.1, the mapping is surjective.

(iii)⇒ (i) Let α be the smallest member of QordKρ (Te(X)) and let F (X) = Te(X)/α ∩ α−1

and F(X) = Te(X)/α = 〈F (X), α/α ∩ α−1〉. F(X) in a member of K and is freely gener-ated over K by { [x]α∩α−1 : x ∈ X }. [x]α∩α−1 and [x′]α∩α−1 are distinct congruence classesif x and x′ are distinct (provided K is nontrivial, i.e., contains a ρ-poalgebra with morethan one element). For convenience we identity { [x]α∩α−1 : x ∈ X } with X.

Fix distinct variables x and y and let β be the ρ-qorder of F(X) generated by {〈x, y〉}.Note that β ∩ β−1 ∈ CoK(F (X)), and that it is a compact element of the lattice CoK(F (X))since it is image of a compact element of QordK

ρ (F(X)). Thus β ∩ β−1 is a finitely generatedAlg(K)-congruence of F (X). Let{ ⟨

[t′i(x, y, z0, . . . , zn−1)]α∩α−1 , [s′i(x, y, z0, . . . , zn−1)]α∩α−1

⟩: i < m

}be a set of generators. Let ti(x, y) = t′i(x, y, x, x, . . . , x), that is, the term obtainedfrom t′i by substituting x for all occurrences of the z0, . . . , zn−1; similarly, let si(x, y) =s′i(x, y, x, x, . . . , x). We will show that { ti(x, y) ≈ si(x, y) : i < m } is a defining set for K.Let A ∈ K and a, b ∈ A. Let X be a set of sufficient cardinality so that there exists anorder epimorphism h from F(X) onto A such that h(x) = h(z0) = · · · = h(zn−1 = a andh(y) = b.

Suppose a ≤A b. Then x h−1(≤A) y. So β ⊆ h−1(≤A). Consequently, β ∩ β−1 ⊆h−1(≤A)∩h−1(≥A), the order kernel of h. So, for each i < m, tAi (a, b) = t′Ai (a, b, a, a, . . . , a) =h([t′i(x, y, x, x, . . . , x)]α∩α−1

)= h

([s′i(x, y, x, x, . . . , x)]α∩α−1

)= sA

i (a, b).Now suppose a �A b. Then 〈x, y〉 /∈ h−1(≤A). So β * h−1(≤A), and hence, since distinct

ρ-qorders have distinct symmetrizations by hypothesis, β ∩ β−1 * h−1(≤A) ∩ h−1(≥A).Thus there is an i < m such that the pair

⟨[t′i(x, y, z0, . . . , zn−1)]α∩α−1 , [s′i(x, y, z0, . . . , zn−1)]α∩α−1

⟩is not in the order kernel of h. Hence

tAi (a, b) = h([t′i(x, y, z0, . . . , zn−1)]α∩α−1

)6= h

([s′i(x, y, z0, . . . , zn−1)]α∩α−1

)= sA

i (a, b).

�

5. Properties of the lattice of ρ-qorders

By a polynomial form (over a signature Σ) we mean a term t(∗, x0, . . . , xn−1) with adistinguished variable, which we denote by ∗, that occurs at exactly one place in t; theother variables, which are assumed to be included in the list x = x0, . . . , xn−1, are calledparametric variables. The set of all polynomial forms is denoted by Pl. A polarity functionρ can be extended from fundamental operations at their various argument positions topolynomial forms in a natural way. ρ(∗) = +. If ρ

(t(∗, x)

)= +, then for any σ ∈ Σm

and i < m, ρ(σ(y0, . . . , yi−1, t(∗, x), yi+1, . . . , yn−1)

)= ρ(σ, i). If ρ

(t(∗, x)

)= −, then

ρ(σ(y0, . . . , yi−1, t(∗, x), yi+1, . . . , yn−1)

)= −ρ(σ, i). The set of all positive and negative

polynomial forms are denoted by Pl+ρ and Pl−ρ , respectively.Let A be a Σ-algebra. A unary function p:A → A on the universe of A is called a

polynomial function over A if there is a polynomial form t(∗, x) and a fixed but arbitraryJanuary 11, 2004


sequence c of elements of A, called the parameters of p, such that, for every a ∈ A, p(a) =tA(a, c). The set of all polynomial functions over A is denoted by Pl(A). Suppose nowthat A is the underlying algebra of a ρ-poalgebra. It is clear that, if t is of positive polarity,then p is monotone, and, if t is negative, p is antimonotone. The sets of all monotone andantimonotone polynomial functions over A are denoted respectively by Pl+ρ (A) and Pl−ρ (A).Let f, f ′ ∈ Pl+ρ (A) and g, g′ ∈ Pl+ρ (A). Then f ◦f ′, g◦g′ ∈ Pl+ρ (A) and f ◦g, g◦f ∈ Pl−ρ (A).

The proof of the following proposition is immediate.

Lemma 5.1. A qordering α of the universe A of the a Σ-algebra A is a ρ-qordering ofA iff, for all a, b ∈ A, aα b implies f(a) α f(a) for every f ∈ Pl+ρ (A) and b α−1 a impliesg(a) α g(b) for every g ∈ Pl−ρ (A).

For any ρ-poalgebra A and set R ⊆ A of pairs of elements of A, we denote by Φρ(R)the smallest ρ-qorder of A that includes R. Note that, if K is the class of all ρ-poalgebras,or more generally any ρ-povariety that contains A, then ΦK

ρ (R) = Φρ(R). We note alsothat Φρ(∅) = ≤A. We obtain a characterization of the ρ-qorder generated by R that isthe natural analogue (and in fact a generalization) of the well-known lemma of Mal’cevcharacterizing congruence generation.

Proposition 5.2. Let A be a ρ-poalgebra and let R ⊆ A2. Let a, b ∈ A. Then aΦρ(R) biff there exists a finite sequence

(19) a = c0, c1, . . . , cn−1, cn = b

such that, for each i < n, one of the following conditions holds.(i) There exists an 〈r, r′〉 ∈ R and an f ∈ Pl+ρ (A) such that ci = f(r) and ci+1 = f(r′),

or(ii) There exists an 〈r, r′〉 ∈ R−1 and an f ∈ Pl−ρ (A) such that ci = f(r) and ci+1 =

f(r′), or(iii) ci ≤A ci+1.

Proof. Let R be the set of all pairs 〈a, b〉 such there is a sequence (19) satisfying the givenconditions. Since ∗ (the identity polynomial form) is contained in Pl+ρ , R ⊆ R. Also itis clear that ≤A⊆ R. To see that R is a ρ-qordering of A, suppose we have a sequence(19) such that, for each i < n, one of the conditions (i)–(iii) holds. Let g ∈ Pl+ρ (A), andconsider the sequence

(20) g(a) = g(c0), g(c1), . . . , g(cn−1), g(cn) = g(b).

Let i < n, and suppose (i) holds for the pair 〈ci, ci+1〉, i.e., there is a pair 〈r, r′〉 ∈ Rand f ∈ Pl+ρ (A) such that ci = f(r) and ci+1 = f(r′). Then g(ci) = (g ◦ f)(r) andg(ci+1) = (g ◦ f)(r′). But g ◦ f ∈ Pl+ρ (A). So (i) also holds for 〈g(ci), g(ci+1〉. If (ii) holdsfor 〈ci, ci+1〉, then a similar argument using the fact that f ∈ Pl−ρ (A) shows that (ii) alsoholds for the pair 〈g(ci), g(ci+1〉. Finally, if 〈ci, ci+1〉 ∈ ≤A, 〈g(ci), g(cI+1)〉 ∈ ≤A since ≤A

is a ρ-qordering. So (iii) also holds for 〈g(ci), g(ci+1〉. Thus 〈g(a), g(b)〉 ∈ R.Suppose now that g ∈ Pl−ρ (A), and consider the sequence

(21) g(b) = g(cn), g(cn−1), . . . , g(c1), g(c0) = g(a).

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24 DON PIGOZZI

Consider any i < n. Arguing as above it is easy to see that if (i), (ii), or (iii) holds for〈ci, ci+1〉, then (ii), (i), or (iii) holds respectively for 〈g(ci+1), g(ci)〉. Thus 〈g(b), g(a)〉 ∈ R.

This shows that R ∈ Qordρ(A) and R ⊆ R. So Φρ(R) ⊆ R. The opposite inclusion isobvious. �

Theorem 2.16 is a corollary of this result.

Proposition 5.3. Let K be a ρ-povariety and let A ∈ K and R ⊆ A2. Then

ΘAlg(K)(R) =(Φρ(R) ∨ Φρ(R−1)

)∩

(Φρ(R) ∨ Φρ(R−1)

)−1.

Proof. Let α = Φρ(R) ∨ Φρ(R−1). Clearly R ⊆ α and also R ⊆ α−1 since R−1 ⊆ α. SoR ⊆ α ∩ α−1, and hence ΘAlg(K)(R) ⊆ α ∩ α−1. By Proposition 4.1 there is a β ∈ Qordρ(A)such that β ∩ β−1 = ΘAlg(K)(R) and obviously R ∪ R−1 ⊆ β. So α ⊆ β and henceα ∩ α−1 ⊆ β ∩ β−1 = ΘAlg(K)(R). �

We now investigate the permutability of ρ-qorderings. In many respects the resultsobtained strongly reflect the theory of permutable congruences, but with some importantdifferences.

Definition 5.4. Let A be a Σ-poalgebra and α, β ρ-qorderings of A. α and β are said topermute if α ; β = β ; α.

A ρ-quasi-povariety K is said to have premutable ρ-qorders if, for each A ∈ K, any pairof K-ρ-qorders of A permute.

Proposition 5.5. Let A be a ρ-poalgebra and let α, β ∈ Qordρ(A). The following areequivalent.

(i) β ; α ⊆ α ; β.(ii) α ∨Qordρ(A) β = α ; β.

Proof. (i) =⇒ (ii). Assume (i) holds. Clearly α ; β ⊆ α∨Qordρ(A) β. Since α∪β ⊆ α ; β,to obtain the inclusion in the opposite direction it suffices to show that α ; β is a ρ-pordering of A. It is reflective because α, β are. It is transitive because (α ; β) ; (α ; β) =α ; β ; α ; β ⊆ α ; α ; β ; β = α ; β. Finally, suppose a α ; β b, and let c ∈ A such thataα c β b. If f ∈ Pl+ρ (A), then f(a) α f(c) β f(b) and hence f(a) α ; β f(b). Now supposea (α ; β)−1b and f ∈ Pl−ρ (A). Then a β−1;α−1 b. Let c ∈ A such that a β−1 c α−1 b. Thenf(a) β f(c) α f(b), and hence f(a) β ; α f(b). So f(a) α ; β f(b) by (i).

(ii) =⇒ (i). β ; α ⊆ α ∨Qordρ(A) β = α ; β. �

It seems unlikely that the inclusion β ; α ⊆ α ; β implies permutability in general,although we know of no counterexample.

We now give an analogue of the well-known Mal’cev criterion of permutable congruences.

Theorem 5.6. Let K be an ρ-quasi-povariety. K has permutable ρ-qorders iff there existsa term p(x, y, z) in three variables such that the following quasi-inidentities hold in K.

(i) x 4 y → p(x, y, z) 4 z.(ii) y 4 z → x 4 p(x, y, z)

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Proof. Suppose that (i) and (ii) hold. Let A ∈ K and let α, β ∈ QordKρ (A). Assume

a α ;β c. Then there exists a b such that aα b β c. Thus pA(a, b, c) α c by (i) anda β pA(a, b, c) by (ii). Hence a β ;α c. So α ; β ⊆ β ; α, and by symmetry β ; α ⊆ α ; β.

Assume now that K has permutable ρ-qorders. Let γ be the smallest ρ-qorder ofQordK

ρ

(Te(x, y, z)

). Recall that Te(x, y, z)/γ is freely generated over K by [x]γ∩γ−1 , [y]γ∩γ−1 ,

and [z]γ∩γ−1 , which we identify respectively with x, y, and z. In Te(x, y, z)/γ we havexΦK

ρ (x, y) ; ΦKρ (y, z) z. So by permutability there is a term p(x, y, z) such that

x ΦKρ (y, z) p(x, y, z) ΦK

ρ (x, y) z.

Let A ∈ K and let a, b, c ∈ A such that a ≤ b. Let h:Te(x, y, z)/γ → A such thath(x) = a, h(y) = b, h(z) = c. The order kernel h−1(≤A) contains 〈x, y〉 and hence includesΦρ(x, y). It follows that pA(a, b, c) ≤ c. So (i) is an quasi-inidentity of K. By a similarargument we get that (ii) is also a quasi-inidentity. �

Any term p(x, y, z) satisfying the condition of this theorem is called an order Mal’cevterm. Suppose p(x, y, z) is a Mal’cev term for Alg(K), i.e., the four inidentities p(x, x, z) 4<z and p(x, z, z) 4< x all hold in K. Then p will be an order Mal’cev term for K providedit has either negative polarity at the y position or positive polarity at both the x and zpositions. Actually, a somewhat stronger property holds, which we now formulate.

Corollary 5.7. Let K be a ρ-quasi-povariety. Assume there is a term p(x, y, z) in threevariables such that the following inidentities hold.

(i) p(x, x, z) 4 z,(ii) x 4 p(x, z, z).

Assume in addition that at least one of the following two conditions hold.(iii) p(x, ∗, z) ∈ Pl−ρ ,(iv) p(∗, y, z), p(x, y, ∗) ∈ Pl+ρ .

Then K has permutable ρ-qorders.

Proof. Assume (i) and (ii) hold. Let A ∈ K and let a, b, c ∈ A. If (iii) holds, then a ≤A

b =⇒ pA(a, b, c) ≤A pA(a, a, c) ≤A c and b ≤A c =⇒ a ≤A pA(a, c, c) ≤A pA(a, b, c).If (iv) holds, then a ≤A b =⇒ pA(a, b, c) ≤A pA(b, b, z) ≤A c, and b ≤A c =⇒ a ≤A

pA(a, b, b) ≤A pA(a, b, c). �

let p(x, y, z) = x · (y → z). In the ρ-povariety of POLRMs we have the inidentitiesp(x, x, z) = x · (x → z) 4 z and x 4 x · 1 4 x · (z → z) = p(x, z, z). Thus POLRM haspermutable ρ-qorders.

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