Category equivalence preserves unification type

Algebra Universalis, 36 (1996) 457-466 0002 5240/96/040457-1051.50 + 0.20/0 �9 1996 Birkhfiuser Verlag, Basel

Category equivalence preserves unification type*

M. H. ALBERT

Abstract. The unification type of a variety is a rough measure of the extent to which systems of equations in the free algebras of the variety have general solutions. We prove, using a concrete characterization of categorically equivalent varieties due to McKenzie, that categorically equivalent varieties have the same unification type.

In August 1992, at the Day memorial conference, Ralph McKenzie presented a

theorem establishing an algebraic criterion for the equivalence as categories of two

varieties U and ~ . We had been searching fruitlessly for some time for examples

of algebraic constructions which preserve the unification type of a variety (see the

definition below) and were quite pleased to have another construction to try.

Somewhat to our surprise we were able to prove that if two varieties are equivalent

as categories then their unification types are the same. The proof is not by "abstract

nonsense" but seems rather to depend on the types of constructions which McKen-

zie showed were sufficient to establish all such equivalences.

The remainder of this paper is devoted to setting the stage for the proof of this result, and then proving it.

1. Definitions and notation

We assume that the reader is familiar with elementary universal algebra with

particular reference to the properties of free algebras in a variety ~ . Either [3] or

[7] would be a more than adequate reference. A nodding acquaintance with the basic terminology of category theory ([5]) will be helpful, but not essential as all the

results can be interpreted in a purely algebraic context. We assume only that the

reader knows what a functor is (roughly speaking a map between categories which is a "homomorphism" of their diagrams.)

Presented by R. McKenzie. Received April 24, 1995; accepted in final form March 4, 1996. *Subject Classifications (1991), Primary 08B20, Secondary 08B05, 08A40.

457

458 M.H. ALBERT ALGEBRA UNIV.

Two varieties U and ~ are equivalent as categories if there is a functor:

F: ~/ --* ~f"

such that for any algebras A and B in "U the induced map:

F: hom~,-(A, B) --* hom~(F(A), F(B))

is a bijection, and for any algebra C in ~ there exists an algebra A in ~U such that F(A) is isomorphic to C. If such a functor exists, then we write:

~ c ~F.

It is of course clear that ~ c is an equivalence relation on the class of all varieties. Two special constructions are known to produce varieties which are equivalent as

categories:

~ "~["] (1)

V ~ c ~(fi). (2)

Here U[n] is a matrix power of ~ (a familiar construction, see [9]). The variety V(r may be somewhat less familiar however. It is determined from ~ and an invertible idempotent unary term a of U , by setting for each algebra A of ~V the algebra A(~) to have universe:

{e cA: a ( a ) = a )

and operations G determined from each term t of "U with interpretation:

t~(y) = ,~(t(y)).

(Notice that we make no notational distinction between elements and tuples, requiring the careful reader to make the appropriate inferences from context.) The invertibility of a means that there exist some terms:

t, tl, t 2 , . �9 �9 , I s

such that:

~!/" ~ X = t ( f i t l ( X ) , f i t 2 ( X ) , . . . f i t s ( X ) ) .

Vol. 36, 1996 Category equivalence preserves unification type 459

Ralph McKenzie has proven ([6]) that the varieties ~ and ~" are equivalent as

categories if and only if there exists a positive integer n and an invertible idempotent term a on ~rnl such that ~ and #-En](a) are term equivalent.

Let ~ be a variety, and letf~ ,f2 . . . . . fk, gl, g2 . . . . . gk be terms of V which we assume to all have the same arity r. We consider the system of equations

A = { f (x ) =gi(x) : 1 < i _< k}.

We wish to find solutions (should they exist) to A in ~ . More precisely, we would like to study the set of substitutions

x, ~ sl (z), x2 ~ s2(z) . . . . . xr ~ st(z)

for which:

~ /~ f~(sl(z) , s2(z) . . . . . s t(z)) =g , ( s , ( z ) , s2(~) . . . . . s~(z)). l_<i<k

A less cumbersome way to identify such solutions is as the homomorphisms:

(where Z is countably infinite), for which:

( f , g~) e ker c~ for 1 < i < k.

Let Sol~(A) denote the set of such homomorphisms. On the set hom(F~(xl , x 2 , . . . , xr), F~(Z)) we have a natural transitive and

reflexive relation _< given by:

c~ < f l ~ there is O:F~(Z)~F~(Z) with O~ =ft .

In other words, thought of as substitutions, fi can be recovered from e by further substitution of terms for the symbols z.

I f c~, fl are both in Solv(A) and c~ _< fl then we say that e is a more general solution of A than ft. We call c~ a most general solution of A if for all solutions fl,

/~_<~ ~ ~ < p .

460 M . H . ALBERT ALGEBRA UNIV.

Of course we have an equivalence relation ~ on Solr(A) given by:

e ~ p <=~ e < / ? and / ? < e

and Sol , - (A)/~ is partially ordered by _<. In this context: e is a most general solution of A if and only if, e / ~ is a minimal element of S o i l ( A ) / ~ .

Henceforth we assume that Sol~(A) is non-empty (or adjust the definitions below so that if it is empty, then A it is considered unitary). We say that:

�9 A is unitary if Solar(A)/~ has a least element, �9 A isfinitary if it is not unitary but every element of Sol,f (A) /~ lies above one

of finitely many minimal elements, �9 A is infinitary if it is not unitary or finitary but every element of Solf(A)/

lies above a minimal element, and �9 A is nullary otherwise.

We informally rank these possibilities in the given order (i.e. unitary is best, nullary worst). The unification type of ~ is the worst of these which occurs for some system of equations A over U.

So the study of the unification type of ~U (or of the fine structure of particular solution sets to particular systems of equations) tells us how well or how badly systems of equations behave. For much more on the general motivation for studying unification type, and many specific examples we shamelessly recommend ([2], [1]). For more general and broad ranging discussions of the unification type problem see [8], and other papers in that volume.

A problem with the unification type is that there are not many general constructions which preserve it, or even affect it in a predictable way. Just for example, the variety of all groups is infinitary, but has both unitary (abelian groups) and nullary (metabelian groups) subvarieties. Also among the varieties generated by clones on a 2-element set there occur unitary, finitary and nullary varieties, with no direct links to the size of the clones (since the naked set, and the primal algebra are both unitary).

So, as remarked above, we were quite excited by the prospect that category equivalence was well-behaved with respect to unification type. In the next section we shall prove:

T H E O R E M 1. Let ~ and ~/U be varieties which are equivalent as categories. Then the unification type of ~U and of ~U is the same.

In fact the proof gives a somewhat stronger (if slightly more technical) result:


T H E O R E M 2. Let r and ~K be varieties which are equivalent as categories.

Then for every system o f equations A over ~U there exists a system of equations A '

over ~ such that:

S o l ~ ( A ) / ~ and Solsr(A')/~,,r .

are isomorphic partially ordered sets.

2. The proofs

From McKenzie's characterization of varieties which are equivalent as categories, it is clear that in order to prove Theorem 1 or Theorem 2 it suffices to show that the two constructions of taking a matrix power of a variety, and taking an

"idempotent projection" of a variety both preserve unification type and/or solution sets (for the second version). That is, to prove Theorem 2 we note that under the stated hypotheses, there is some n and some idempotent r such that V = ~[-l(z)

(McKenzie 's result). So to prove this result we need consider only the two cases = ~/U E'] and ~U = ~#F(o-). This is precisely what we do in Propositions 3 and 4

below. We use ~ to denote isomorphism. The easier case is the matrix power.

P R O P O S I T I O N 3. Let ~U be a variety and n a positive integer. Then for any

system o f equations A over ~ there exists a system of equations A ['q over ~/~["] such that:

S o l ~ (A) / ~ ~ ~ Sol~[,,] (A t,,]) / ~ - r [ , , l -

Conversely, given any system o f equations A ['] over "!/~[~] there ex&ts a system A over such that the above isomorphism holds.

Proof. Both parts of the proposition follow almost immediately from the observation that there is a one to one correspondence between k-ary terms in ~-[-1

and n-tuples of kn-ary terms in ~ , and that equality between such terms occurs if and only if each component of the corresponding n-tuples are equal.

So given a system of equations A over ~ , pad the list of variables to have length kn for some k, add extra equations stating that all the new variables equal the first of the original variables, and then pad the list of equations by duplicating equations so that the number of equations is also a multiple of n. Obviously this new system has a solution set isomorphic to the original one. Now each block of n equations


gives a single fL'Lequation in k variables, and again the solution sets are clearly isomorphic.

In the reverse direction just replace each of the ~ ' 1 equations in k variables by the corresponding n equations in kn variables over ~U.

Now suppose that o- is an invertible idempotent unary term for ~/, and fix:

t, t l , tz, . . . , t s

such that:

~ x = t ( a q ( x ) , a t a ( x ) , . . , ats(x)).

The connection between the free algebras in ~ and ,4r(a) works as follows (see McKenzie's paper for the details). Suppose that F ( X ) is the free algbra in generated by X. Let A ( X ) be the subalgebra (in Y/) of F ( X ) generated by the elements ~r(x) for x in X. Then A ( X ) ( a ) is (naturally isomorphic to) the free algebra in V(a) freely generated by the elements a(x) for x in X (i.e. it has the same rank as the free algebra F ( X ) . )

We will show how to associate to each system A(a) of f ( a ) equations, a system A of Y/~-equations such that:

Sol~ (A)/~ ~ ~ Sol~(~)(A(o'))/~ ~<~).

Specifically, suppose that A(a) is a system of equations in

X : X l ~ X 2 ~ . . . ~ X n .

Then for each equation

L ( x ) = g~(x)

in A(o-) we include the equation:

a ( f ( x ) ) = ~(g(x))

in A, and we also add the equations:

~r(xi) = xi for 1 _< i < n

to A. This gives a complete description of A.


PROPOSITION 4. Le t 7s be a variety, and let a be an invertible, idempotent

term o f ~U. Then f o r each sys tem o f r equations, A(a) there exists a sys tem A o f

equations (whose construction is described above) such that:

Solr-(A) / ~ ~- ~- Sol~;(~) (A(a)) / ~,~(~).

P r o o f Let Z be a countably infinite set, F ( Z ) the free algebra in ~ generated by Z, let X be the set of variables in A, and let A ( X ) < F ( X ) and A ( Z ) < F ( Z ) be as described above. We adopt the convention that any modification of z stands for an element of Z, and ones which appear different are different (for example Zl, zll and z21 stand for distinct elements of Z.) We will call a solution:

~: F ( X ) -* F ( Z )

of A "good" if

c~(F(X)) < A ( Z ) .

CLAIM 5. For every solution ~ o f A there exists a good solution a' o f A such that

O C ~ I.

P r o o f Suppose

~(xi) = ri (z , , z 2 , . . . , zk).

Then since A includes the equation

(Y(Xi ) = X i.

we must have:

r~(z~ , z2 . . . . . ~k ) = ~ r i ( z l , ~ . . . . , z D .

Define:

a'(xi) = ari ( t(aZll , . . . , azl~), t(az21 . . . . , az2~ ), t (azkl , . . . , az~)) .


Then plainly e < c~' via any endomorphism of F(Z) satisfying:

zj ~ t(~zjl . . . . . ~zj,) .

for 1 < j < k. But also ~' < ~ via any endomorphism of F(Z) satisfying:

zij ~-~ atj(zi). [] (of claim).

So we may restrict our attention to good solutions, and henceforth we shall do so. The main reason for doing this is that to a good solution ~ for A, we can associate a homomorphism ~(a) between the free algebras in U(a ) which is a

solution of A(a), and this allows us to compare the solution sets

Again since

~ ( x , ) = xi

is an equation of A, we have that for any good solution ~,

O~(CTXi) = O~(X i) .

In other words there is a one to one correspondence between good solutions and

their restrictions to A(X). But from McKenzie's result we know that:

hom~-( A( X), A( Z ) ) ~- hom,-(~) ( A( X)( a), A( Z )( a) ),

and the algebras on the right are free in ~'(a). Since the isomorphism from left to right is just given by restriction then it is clear from the construction of A that the homomorphism from A(X)(a) to A(Z)(a) corresponding to a good solution of A will be a solution of A(a). So, provided that the restriction of good solutions-to A(X) preserves < ~ we will obtain the desired isomorphism between

Sol~-(A)/~ and Sol~:~(A(a))/~,~G).

(We should really argue that extension also preserves _<, but this is immediate.) So suppose that c~1 and 0~ 2 are good solutions of A with:

~i (x j ) = Gs~(~z~ . . . . . ~z~),

for i = 1 , 2 and l_<j_<n and that

O: F(Z) ~ F(Z)


is such that

00{ 1 = ~ 2 o

Suppose that:

O(z,) =p,(z),

and define

6(~,) = op,(~z),

and on any other generators define O(z) = oz. Since 0el = ~2,

OOSli(OZ1 . . . . . O Z k ) = f f S 2 / ( O Z l , . . . , O 'Zk)

for 1 < i _< n. That is:

V ~ OSli(ffpl ( z ) . . . . . f f p k ( z ) ) = O S 2 / ( O Z 1 , . . . , OZk).

But then, replacing all z's by az's we get:

V ~ OSli(Opl (OZ) . . . . . opk(az)) ~-- OS2i(02Z1 . . . . . 02Zk) = OS2i(OZ 1 . . . . . OZk).

In other words 0~i = e2. But the range of Ois contained in A(Z) so its restriction to A(Z) and thence to A(Z)(a) witnesses that

O{ 1 ( O ) ~ ~ 2 ( 0 ) in ~(~r).

As noted above, extensions of solutions of A(o) to hom(A(X), A(Z)) and thence to good solutions clearly preserves _< so we have established the desired isomorphism. []

3. Applications

By noting that the variety of Boolean algebras is unitary, and that any two varieties generated by primal algebras are equivalent as categories, it follows that any such variety is unitary (of course this is also easy to establish by elementary means, see [4]).

466 M.H. ALBERT ALGEBRA UNIV.

For a finite ordered set P, let Y/~(P) be the variety generated by the algebra with underlying set P together with all order preserving maps P" to P (all n). I f P is a lattice then there is a retraction from 2 x to P for some X (here 2 denotes the two

element chain) and hence ~ ( P ) is equivalent as a category to the variety of bounded distributive lattices. Thus the unification type of the variety ~ ( P ) is the same as that of the variety of distributive lattices, i.e, nullary (a proof that the

variety of distributive lattices is nullary, a restdt due to R. Willard, can be found in [2].) In fact, with a little extra work one can prove the same result for finite partial

orders in which every non-empty set which has an upper bound has a supremum.

Acknowledgements

We would like to thank the organizational committee of the Day memorial

conference, especially M. Valeriote, for their outstanding work. Also Ross Willard who kept the author from crashing the car between Hamilton and Waterloo while excitedly expounding these ideas. We also acknowledge the assistance of an anonymous referee who found many typographical errors (those remaining are the

sole responsibility of the author) and suggested a number of improvements in the

presentation.

REFERENCES

[1] ALBERT, M. H. and LAWRENCE, J., Unification in varieties of groups I: nilpotent groups. Canadian Journal of Mathematics, 46 (1994), 1135-1149.

[2] ALBERT, M. H. and WILLARD, R., Unification in locally finite varieties, preprint 1992. [3] BURRIS, S. and SANKAPPANAVAR, H. P., d Course in Universal Algebra, Springer-Verlag, New

York, 1981. [4] NIPKOW, T., Unification in primal algebras, their powers and their varieties, J.A.C.M., 37 (1990),

742-776. [5] MAcLANE, S., Categories for the Working Mathematician, Springer-Verlag, New York, 1971. [6] MCKENZIE, R., Algebraic Morita theorem for varieties, preprint, August 1992. [7] MCKENZlE, R., McNULTY, O. and TAYLOR, W., Algebras, Lattices, Varieties, Wadsworth/Brooks-

Cole, Monterrey, 1987. [8] SIEKMANN, J., Unification theory, J. Symbolic Computation, 7 (1989), 207-274, [9] TAYLOR, W., Thefine spectrum of a variety, Algebra Universalis, 5 (1975), 263 303.

Department of Mathematics University of Otago Dunedin, New Zealand

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Category equivalence preserves unification type