Algebra and Logic, Vol. 35, No. 1, 1996

T H E S T R U C T U R E O F C O N G R U E N C E L A T T I C E S O F F I N I T E S E M I L A T T I C E S

K . V . A d a r i c h e v a UDC 512.567.3

G. ghitomirskii gave a description of congruence lattices of semilattices in the second-order

language. Here we describe finite lattices belonging to this class in terms of properties shared by

coatoms (minimal nonidentity elements of a lattice}. As a consequence, a characterization of

finite semiIattices with isomorphic congruence lattices is obtained.

I N T R O D U C T I O N

The class K of congruence lattices of semilattices has long been the subject of investigation. In [1], a

number of necessary elementary properties of these lattices were established. (Such lattices, for instance,

are meet semidistributive, generated by coatoms, and relatively complemented.) These properties, however,

did not suffice to describe the class K even in the finite case. In [2], it was shown that every nontrivial

lattice identity fails in the congruence lattice of some finite semilattice. In [3], it was proved that K is not

elementary axiomatizable, and a description of this class in the second-order language was offered. The

main result of [3] states that if a complete lattice contains a subsemilattice P with certain properties, then

it can be represented as a lattice Con(P) of congruences of P. An elementary class of lattices embeddable

in lattices from K was described in [4]. Distributive and Boolean lattices in K were studied in [4-6]. Current interest in the class K! of finite lattices from K has arisen primarily because these proved to

be upper bounded. Recall (see [7]) that a lattice L is upper (lower) bounded if there exists a homomorphism

from a finitely generated free lattice onto L, under which the preimage of any element of L has a greatest

(least) element. It is known that any upper bounded lattice is meet semidistributive. Therefore, the

upperboundedness is a better tool for describing KI, just as the Arguesian law is thought of as best

reflecting the nature of lattices of normal subgroups in the class of modular lattices. In this paper we use

the property of being upper bounded to describe the class K I.

In fact, we will deal with lattices dual to congruence lattices of semilattices. A partial upper subsemi-

lattice of a finite lower semLlattice with zero 0p is a subset X C P that contains 0p nnd is dosed under the

partial binary operation + of taking the least upper bound. Partial upper subsemilattices P form a lattice,

with respect to inclusion, denoted Sub+(P). We approach the description of K! using the following:

L E M M A [2]. The congruence lattice of a finite lower semilattice P is dually isomorphic to Sub+ (P).

Note that if a semilattice P has a greatest element 1p, then the lattice Sub+ (P) coincides with Subv (P),

the lattice of upper subsemiJattices of P with Op. Lattices of the form Subv (P) were described in [9], where

it was shown that the property of having no special sequences of atoms, called cycles, is equivalent to being

lower bounded (see also [10, 11]). It was noted in [8] that lattices Sub+(P) satisfy a similar condition, and

Translated from Algebra i Logika, Vol. 35, No. 1, pp. 3-30, J~tnuary-Febru~ry, 1996. Original article submitted July 4, 1994.

0002-5232/96/3501-0001 $12.50 (~) 1996 Plenum Publishing Corporation 1

hence lattices of the form Con(P) are upper bounded. We believe that the technique developed to describe

Subv(P) in [9] is by and large applicable to congruence lattices of finite semilattices.

Indeed, the description of lattices of the form Sub+(P) given in Theorem 2.6 continues the character-

ization of Subv (P) from [9] (see Theorem 1.3 below). All properties specified in the theorems are sort of

conditions on atoms, which can be verified more effectively than those in [3]. This problem is discussed

in detail in Section 4. Unlike lattices Subv(P) , however, not all the properties describing Sub+(P ) can

be expressed in the first-order language. For this reason, the question remains open as to whether K! is

elementary axiomatizable relative to the class of finite lattices. It is worth noting tha t the class S ( K I ) of

lattices embedded in finite congruence lattices is exactly one consisting of finite upper bounded lattices [12,

13]; hence, it is elementary axiomatizable relative to the class of finite lattices. Is the class S(K) of lattices

embedded in lattices from K dementa ry axiomatizable?

An important consequence of our description is getting an idea of how associate semilattices, i.e. semi-

lattices with isomorphic congruence lattices, are structured. Here we can draw the analogy with the problem

of describing semilattices having isomorphic subsemilattice lattices. For the generai case, this problem was

solved in [14]; for finite lattices, the result follows from the description of lattices of the form Subv (P) given

in [9]. Using the characterization of Sub+(P) , we can similarly resolve the association problem for finite

semilattices, and do this in Sec. 3.

For the lattice notions unspecified in the paper, we ask the reader to consult [15].

N E C E S S A R Y C O N D I T I O N S F O R A L A T T I C E T O B E P R E S E N T E D AS Sub+(P )

Let P be a finite semilattice with the least element Op. Clearly, Sub+(P) is an atomistic lattice with

a toms of the form ~Op, z~ for an arbi trary nonzero element z E P. Recall tha t a lattice is said to be

a~omis~ic if every element in this lattice is a join of atoms. For convenience, we will not distinguish between

a toms of the lattice L = Sub+(P) and elements of the semilattice P, e.g., an a tom {0p, x} is denoted simply

by z. Evidently, the join a v b of two arbi trary atoms, a and b, in the lattice Sub+(P) contains at most

three atoms. To denote this property of atomistic lattices, write (D2). It is also clear that if elements a

and b of a semilattice P either are comparable or do not have an upper bound in common, then the join of

the corresponding atoms a and b in L contains at most two atoms (including the case a -- b). But if a and

b have a common upper bound in P, they also will share the least upper bound c = a + b, which means

tha t c <_ a v b in the lattice L.

D e f i n i t i o n 1.1. Let L be a lattice satisfying condition (D2) and let At(L) be the set of a toms in L. By

a socle of L we mean a system IP = (At; o,~-.), where o is a partial binary operation defined for a, b E At(L)

such that a V b contains exactly three atoms, and a o b = b o a = c for c _< a v b, c ~ a,b; -~ is a binary

relation consisting of the pairs (a, b) and (b, a) for which a V b contains at most two atoms.

We note that if P is a lower semilattice with the greatest element, and L -- Sub+(P) , then the relation

is satisfied only by those elements a, b of the socle ]P of L which are comparable in P. Recall (cf. [9])

tha t a sec o,, of e A t ( L ) is a set X = U {t A t ( L ) / � 9 = o o ( . . . o t ) . . . ) ) ) for

some t l , . . .~,~ E At(L) and n E N). The sectors of elements z, y, z will be denoted by the corresponding

upper-case letters X, Y, Z. Moreover, a subset U of At(L) is a sector if tr is a sector of some element

u E At(L), and we then call u the bottom of U. It should be observed that elements of a sector, when

considered as elements of P, form a convex subset in P (see [9]). An atomistic lattice L is said to be

bia~omic if, for any a tom c and all a, b in L, c <__ a V b implies c _< a t V b ~ for some a toms a ~ < a and b ~ <_ b.

A cycle is a sequence of atoms z0, z l , . . . , z , ~ = zo, n > 2, such that zi = zi+x o ti for some ti G At(L),

i = O, n - 1. For a biatomic lattice, this definition coincides with the well-known notion of a C-cycle (see

[10, l l D. For our further reasoning, we need the following:

L E M M A 1.2 (see [16]). Let L be a finite biatomic lattice which satisfies (D2) and has no cycles and

let ]P be its socle. Then the lattice S(]?) of subsystems of 1? is isomorphic to L. Specifically, if the socles 171

and ~2 of two such lattices L1 and L2 are isomorphic, then LI ~ L2.

P r o o f . Define a mapping ~0: L --* S(]P) by putting ~o(z) = {a E A t ( L ) / a < z}. Since L is atomistic,

~, is one-to-one. Evidently, ~,(z. y) = ~0(z) n ~0(y), and the biatomicity implies ~0(z V y) = ~o(z) V ~o(y). If

X E S(]P) and z = V X , then the biatomicity implies ~0(z) = X, and so ~o is onto.

Recall the definitions of a descent and a zigzag given in [91. Let ~ = (At(L); o; --.) be the socle of a

lattice L. A sequence of pairs of atoms (al, b), (az, b ) , . . . , (a,,, b) is called a left descent in I? if the following

are satisfied: (1) al "-- b; (2) ai+x = a~ o t~ for some ti q At(g); (3) if t E At(L) and b ~ t, then a , --. t. tx t3 t , , - t

Sometimes, to denote a left descent, we write (a~, b ) / z (a2, b ) / . . . / (a,,, b) with elements t l , . . . , t , ,_ l

from the definition. A sequence (b, al), (b, a2) , . . . , (b,a,,) satisfying conditions (1)-(3) is called a r/ght tl t2 t,~--I

de~en~ in ~, and we write (b,~A \ (b ,~) \ . . . \ ( b , ~ ) if necessary. ~ote that ~ 1 , t ~ , . . . , ~ _ l , t ~ _ ~ belong to the sector of a~. We say that the socle of L satisfies the unique descen~ condition (UDC) if any

two left descents beginning with a common pair will end in some common pair as well. Clearly, the same

is true for right descents.

A sequence of atom pairs is called a zigzag in I~ if it consists of alternating right and left descents such

that a last pair of each preceding descent begins the next, and the last pair (u, v) of that sequence satisfies

the condition that, for any t E At(/5), u ~ t i f f v ~ t. A zigzag is even (odd) if the number of descents of

which it is composed is even (odd). A lef2 (right) zigzag is one that begins with a left (right) descent. A

zigzag is called ezact if it ends in a pair (u, u). We say that the socle ~ of a lattice L satisfies the zigzag

condition (ZC) if any exact left and right zigzags beginning with a common pair have different parities.

We proceed to formulate a theorem on the structure of finite subsemilattice lattices.

T H E O R E M 1.3 [9]. A finite atomistic lattice L can be represented as the lattice Subv(P) of sub-

semilattices of P, an (upper) semilattice, iff g is biatomic, satisfies (D2), contains no cycles, and its socle

satisfies UDC and ZC.

In view of the remarks made above, this result my well be tailored for describing lattices of partial upper

subsemilattices of lower semilattices with a greatest element. We will verify which of the above-specified

conditions are satisfied in the lattice Sub+(P) for an arbitrary finite semilattice P . As was noted above,

Sub+(P) satisfies (D2). It is easy to verify that, for any A, B E Sub+(P) , their join in this lattice has the

form A V B : {a + b/, a E A, b E B, and a,b have a common upper bound in P}. Therefore, Sub+(P)

is biatomic. If a o b = c for some atoms a, b, and c in Sub+(P), then a and b are incomparable in P, and

a + b = c, whence a, b < c, a, b ~ c. Thus Sub+(P) cannot have cycles.

Verification that the socle of Sub+ (P) satisfies ZC requires a more sophisticated argument. Suppose, for t l t3

instance, that an exact left zigzag beginning with (al, bl) consists of three descents: (al, bl) 7 (a~., b x ) / z

... / (~,,, b~) \ (a~, b~) \ (~,,, b~) \ ... \ (a~, ~ ) / (a~+~, ~ ) / (~,,+~, ~ ) / . . . / (~,,+,,,, b~) = (w, w). By definition, all the elements involved in forming pairs for this zigzag belong to the sector of w. In

particular, they all have a common upper bound w in P, and hence the relation --, is satisfied only by those

elements which are comparable in P. Since a , < a,,+,, = b~ = w and a~ ~ s ~ _ ~ , . . . , c~ .~ s~, a , ~ Sl,

a,, .~ b~, we have a,, < b~_~,..., a , < b~, and a , < bl. Further, since b~ ... t , _ ~ , . . . ,b~ -~ t2, and b~ -~ Q, it

follows that a,,_~ < b~, . . . , a~ < b~, and a~ < b~. Thus, the fact that the exact zigzag starting with (ax, b~)

terminates at a left descent implies that ai < bi in P. It is easy to verify that our argument does not depend

on the number of descents of which a zigzag is composed. Thus, if an exact zigzag (al, b l ) , . . . , (v, v) starts

with a right descent, it must contain an even number of alternating descents to terminate at a left one.

This means that ZC is satisfied in the socle of Sub+(P) (cf. the proof for Subv(P) in [9]).

P r o p o s i t i o n 1.4. If P is a finite lower semilattice, then L = Sub+(P), the lattice of its partial upper

subsemilattices, is biatomic, satisfies (D2), has no cycles, and its socle satisfies ZC.

C O R O L L A R Y 1.5 (cf. [8]). For every finite semilattice P, the lattice Con(P) is upper bounded.

P roof . It is sufficient to show that the lattice Sub+(P) is lower bounded. Since Sub+(P) is biatomic

and has no cycles, it is freed of C-cydes. In [11], it was shown that for a finite lattice, the latter condition

is equivalent to being lower bounded.

Note that the unique descent condition is not included in the list of necessary conditions for Sub+ (P).

An example showing that the socle of a lattice Sub+(P) may fail to satisfy UDC is as follows. Let Pi be

a semilattice shown in Fig. 1. In the socle ~i of the lattice Sub+(Pi), we then have two right descents Cl C2

(a, b) "~ (a, dl) and (a, b) ~ (a, d2) beginning with (a, b) and ending in different pairs.

Although UDC does not hold in lattices of the form Sub+(P) in general, we are able to define its local

analog. Namely, let X C_ At(L) be a sector in a lattice L with the socle ~. A sequence of atom pairs

(ai, b), (az, b ) , . . . , (a , , b) w i l l be called a descent in ~ relative to X , or an X-descent, if: (1) ai ~ b; (2) al ,

a 2 , . . . , a,,, b E X; (3) ai+x = ai oti for some ti E X , ti "~ b; (4) if t E X and t .~ b, then t ~ a~. Any descent in 11 r with elements of pairs from X is, obviously, an X-descent. Therefore, the bottom z of this sector is

a common upper bound for arbitrary a, b E X. In particular, if a ~ b, then a and b are comparable in P.

Using this remark and repeating the proof of the corresponding assertion for lattices of the form Subv(P)

from [9], we obtain the following:

P r o p o s i t i o n 1.6. Suppose that P is a finite semi.lattice, t? is the socle of Sub+(P), and X is a sector

in At(L). Then ~ satisfies the unique X-descent condition, i.e., a* = a** whenever (a, b ) , . . . , (a*, b) and

(a, b ) , . . . , (a**, b) are X-descents.

Propositions 1.4-1.6 make it possible to distinguish the maximal fragments in L = Sub+(P) represented

by subsemilattice lattices. We call a sector X of a lattice L limit if it is not a proper subset of any other

sector. Accordingly, a sublattice L1 = (X) of a lattice L is said to be limit if it is generated by a limit

sector X C_ At(L). It is easy to check that any limit sublattice is an ideal sublattice; namely, L1 = {a E L~

a < VX}. Theorem 1.3 can be combined with the propositions proved above to yield

P r o p o s i t i o n 1.7. Every limit sublattice Li of the lattice L = Sub+(P) can be represented as t h e

subsemilattice lattice of a suitable finite semilattice.

In fact, if Li < L is limit, then it is generated by the limit sector X. Obviously, X is an upper semilattice

and L1 ~ Subv(X).

di ~ d2

Cl C 2

el

Fig. I

S T R U C T U R E OF L A T T I C E S O F T H E F O R M Sub+(P)

In this section, we give sufficient conditions for representing a finite atomistic lattice as a lattice of

partial upper subsemilattices of some lower semilattice. This, together with Proposition 1.4, helps us

furnish a description of these lattices similar to one given in Theorem 1.3 for lattices of subsemilattices. In

what follows we consider the various semilattices defined on the same set P. Write Sub+(P, <:) in place of

Sub+(P), specifying the partial order _< on a semilattice implied. First, we show that, for any semilattice

(P; _<), there exists a "weakest" order (i.e., one containing a least number of pairs of elements from P) in the

set of partial orders < ' C < such that Sub+(P, _<') ~ Sub+(P, <). In other words, we distinguish those pairs

of elements from P whose comparability is implied by the structure of Sub+(P) . For an atomistic lattice

satisfying (D2), we then introduce the notion of a supersocle the definition of which is based on the "weak"

order on P, described above. The main result of the paper is Theorem 2.6 in which the notion of a supersocle

plays an essential role. At the end of the section, we will establish certain conditions distinguishing lattices

of subsemilattices in the class of lattices of the form Sub+(P) and show how Theorem 1.3 can be derived

from Theorem 2.6.

Consider an atomistic lattice L with the socle ~ : (At(L); o; ~). If elements a and b belong to some

sector X _C At(L), we say that they are conjugate by the sector X or just conjugate. Write --~ (a, S), where

S C_ At(L) a n d a E A t ( L ) - S , i f a . ~ s for any s E S. For a, b ,c E At(L), the pair (a,b) is said to be

obtained by a fan shift from (a,c) if there exists a sequence of atoms co = c, c t , . . . ,c,~ = b, called a fan sequence, such that el, ci+l are conjugate by a certain sector Si for which --. (a, Si), i : 0, n - 1. We say

that S __ At(L) ~ is closed with respec~ ~o a fan shift if (a, b) E S whenever (a, b) is obtained by the fan shift

from (a, c), and (a, c) E S. For any S C At(L) "~, denote by S* the least subset in At(L) 2 which contains S

and is closed with respect to a fan shift.

Now let (P; _<) be a lower semilattice, L = Sub+(P), and ~ -- (At; o;..~) be the socle of L. Consider the

relation

<c -- (_< N{(a, b) E At(L)2/a, b are conjugate}) U {(0e, z ) / z E P}.

Note that if (a, b) E ~c, then the elements a and b are comparable in any semilattice (P; <C') for which

Sub+(P, < ' ) ~ Sub+(P, _<). Consider the closure <* of the relation _<c with respect to a fan shift.

L E M M A 2.1. If Sub+(P,_<') -~ Sub+(P, <) and <' C _<, then <C; C_ <' .

P r o o f . It suffices to verify that the relation < is closed with respect to a fan shift, i.e., that if a < c, a,

c are conjugate, and b, c are conjugate by a sector S with -.. (a, S), then a <_ b. For the bot tom s of S we

have s > c > a, and so a is comparable with all elements from S. But if a _> t for some t E S, we obtain

a E S because every sector in P is convex, a contradiction.

The notions introduced are illustrated by the semilattice P2 given in Fig. 2. Obviously, '<c -- _< - { ( z , y),

(a, cl), (a, c2), (a, st), (a, s2)}. We verify that (a, cl), (a, c2), (a, sl), and (a, s2) are obtained by the fan

shift from (a, co). Indeed, a, co G So, st,co, el E'S1, s2,ct,c2 E $2, and also ~ (a, S1), --~ (a, S2). Thus, the

pairs (a, cI), (a, cz), (a, sl), and (a, s2) belong to _<;. Hence _<; = < - { ( z , y)}.

Below we show that the relation _<~ does induce the semilattice structure on the set P and, moreover,

this semflattice is associated with the original semilattice (P; <_), i.e., Sub+(P, <_~) - Sub+(P, _<). We prove

this in the foUowing three lemmas.

L E M M A 2.2. For any semilattice (P; _<), the relation _<~ is a partial order on P .

P r o o f . It suffices to verify that _<~ is transitive. If a < b and a and b are conjugate, it will be convenient

to regard (a, b) to be obtained from (a, b) by the fan shift consisting of a one-dement fan sequence. Let

sz sl so ~/

0

Fig. 2

a <c b -<c c. We can assume that a ~ 0p and that a and c are not conjugate. Let (a, b) be obtained from

a conjugate pair a < d by the fan shift t0 : d, t l , . . . , t k - 1 = b, and (b,c) be obtained from a conjugate

pair b < e by the fan shift so = e, 8 1 , . . . , 8 , = c; in addition, b, e E U0 and 8i-1, 8~ E U~, ... (b ,U~) ,

i = 1 , . . . , n , where Uo , . . . , U,, are some sectors. If a E Uo, then (a,e) E<__~, whence also a • Ui, since

otherwise a < b < 8~ E U~ implies b E U~, which it should not. Consequently, ~- (a, U~), and (a,c) is

obtained from (a, e) by the fan shift 80 = e, 81 , . . . , 8,, = c. If a r Uo, we have ~. (a, Uo) since a _< b for

b E Uo. Then (a,c) is obtained from (a,d) by the fan shift vo = d, vl = t ~ , . . . , v k - 1 = t k - 1 - b, vk = 80 = e,

1;k+l = Sl, �9 .., 1)&+n --- Sn = C.

LEMMA 2.3. If0p~a<~ banda<d<b, thena<~d_<~ b.

Proof. If a and b are conjugate, this is obvious. Suppose (a, b) is obtained by a fan shift from the pair of

conjugate elements ( a , t ) , a < t . Letting a, t q So, assume that there exists a sequence to -- t, t l , . . . ,tk : b

of elements from P in which t j -1 , tj are conjugate by a sector Sj, and ~ (a, Sj) , 1 < j < /c. Consider d

and the sector Sk with elements b : tk and tk-1. If either d ~ 8, or d is incomparable with 8 for some

s E Sk, then d E Sk because d < b < 8k, where s~ is the bottom of Sk. Therefore, d <~ b because these are

conjugate by the sector Sk, and a <~ d since this pair is obtained from (a, t) by fan shifting. We consider

the case where d < 8 for aLl s E Sk and, in particular, ~ (d, S/~). We have d < tk-1, where tk-1 E Sk fSSk-1.

For this reason, if either d ~ s, or d is incomparable with s for some 8 E Sk-1, we have d E S~_1 and

a <~ d <~ tk-1 by analogy with the previous case. Since tk-1, tk are conjugate by the sector Sk for which

-~ (d,S~), the last inequality yields d <~ tk : b by definition. Thus, we have d < 8 for all 8 E Sk-1, and

-~ (d, Sk-1) in particular. Arguing in a similar way, we conclude that either a _<~ d <_~ tj for some j < k

such that ~ (d, S~), . . . , ~ (d, S~+~), whence a <~ d <_~ t~ by definition, or d < 8 for all 8 E S ~ . . . ~ S ~ [in

paxticulax, ~ (d, S1) . . . . , ~ (d, S~)]. We show that d ~ So in the latter case. Since to ~ S~ ~ So, we have

d _ to. Since a _< d and a and t0 axe conjugate by the sector So, we have d q So because So is convex. So

a _<~ d _<~ to, which combines with ... (d, S~) , . . . , -~ (d, S~) to yield d _<~ b.

L E M M A 2.4. <P; _<~) is a lower semilattice, and Sub+(P, <~) ~- Sub+(P, <).

P r o o f . Recall that _<~ C <. We will show that any dements a, b q P with a common upper _<~-bound

have the least upper ___~-bound which coincides with the least upper <-bound. For (a, b) E <~, this is trivial.

Let a and b be incompaxable in (P; _<~) and a, b _<~ d. I f a and b are comparable in (P; <__> and, for instance

a < b, then from a <~ b < d and a _<~ d we obtain a _<~ b by Lemma 2.3. This is a contradiction, which

proves that a and b a~e incompaxable in (P; <_>, where they have a common upper bound d. Consequently,

d' = a + b exists. Since a and d' axe conjugate, a _<~ d', and analogously b _<~ d', from which we infer that <* d' is the least upper bound for a and b in (P; <~). Since 0~ is the least element in (P; _~>, it is a lower

semilattice.

We show that Snb+(P, _<) ~ Sub+(P, _<~). By Lemma 1.2, it suffices to verify that the socles 1t ~ --

(P; o, -~) and ~1 = (P; ol, ~1) of these lattices are isomorphic. From the argument given above, it follows

that a o b : d i f f a 0 1 b : d . Hence also ( a - - . b C ~ a ~ l b ) for a n y a , b E P .

We proceed to find some conditions for a socle of an atomistic lattice L which specify that a semilattice

P with Sub+(P) ~ L is determined on the universe of the socle. In view of the lemmas proved above,

the order ~ on this semilattice can be thought of as weak, i.e., ~ = ~ : . Since _~* is induced by pairs of

conjugate elements, we need only order the elements belonging to the sector. By Proposition 1.7, elements

of the sector X generate a sublattice L1 in L which is isomorphic to Subv(X), where X is considered as an

upper semilattice. Note that the socle of the lattice L1 is a subsystem of the socle of L, and.so the algorithm

suggested in [9] can well be used to order the elements a, b E X, a ~ b. We will recall the main steps of

that algorithm, modifying some of the definitions by introducing local analogs relative to the sector X.

Specifically, we call a sequence (a, b),. . . , (u, v) an X-zigzag if it consists of alternating X-descents, and

for all ~ E X, u --~ t r v -'- t. A pair (a, b) E X 2 is called X-free if, for any t G X, the following conditions

are satisfied: (1) a ..~ t r b ~ t; (2) A ~ t r B ..~ ~ for the sectors A and B of elements a and b,

respectively. Here we have U ~ t for U C_ X, t E X, which means that u' ~ t for some u' e U. (In [9], a

pair (a, b) E X - At(L) satisfying (1) and (2) was called completely free.) If a, b G X, a r b, and a ~ b,

then a and b must be ordered. Since L1 has no cycles, there exists at least one X-zigzag beginning with

the pair (a, b), say, (a, b ) , . . . , (u, v). We then put a < b if one of the following is met:

(a) u = v and the zigzag ends in a left X-descent;

(b) there exists ~ G X such that U ~ $ and (~, v') ~ ~ for all v' E V.

UDC and ZC ensure that such ordering is well defined. If none of the conditions (a), (b) is met, then

the pair (u, v) is X-free. According to [9], elements of fre e pairs can be ordered in an arbitrary way, and

that ordering will then have to be preserved by all elements of the sectors U and V: if u _< v, a E U, and

b E V, then a < b. Every ordering of X-free pairs results in producing a new semilattice with a given lattice

of subsemilattices L1. Moreover, every semilattice X ' with Subv (X t) --- L1 can be obtained in just this

way (for more details, see [9]). In the case where X is a part of P, the semilattice under construction, we

can find it necessary to impose restrictions on the types of ordering of X-free pairs. First, the elements u,

v of an X-free pair may at the same time belong to another sector, from which they inherit the ordering.

Second, even if the pair (a, b) is X-free for every sector X, containing the elements a and b (such pairs are

called free), it does not always admit an arbitrary ordering.

Consider the following example. Let P3 be the lower semilattice given in Fig. 3, and L = Sub+(P3).

The sets X ~ ~ , d , a , c , f , g } and Y -- ~y,a, b,e} are the limit sectors. According to Proposition 1.7, we

can define on X and Y the upper semilattices (X; <) and (Y; ~) such that Subv(X) ~ LI = (X) _< L and

Subv(Y) --- L2 -- (Y) _< L. It is clear that c < a in (X ;~ ) since the sequence (c,a) is an X-zigzag, d ~. C,

and ~.d, a) ~ , i.e., condition (b) is satisfied. The pair (a, b) is free since it is Y-free, and Y is the only

sector containing a and b. Thus, L~ ~ Subv (Y') ~ Subv (Y ' ) , where Y' and yH are the semilattices given

in Fig. 4, which differ by the orderings of a and b. However, the case b <t a cannot be realized in any

semilattice (P3;<'~ ~rith Sub+(P3, <') -~ Sub+(P3, <). Indeed, since b .-. c, it follows that b and c either

are comparable in (P3; ~ ' ) or have no common upper bound in (P3; _<'). Since c ~ ' a, we have b < ' a,

from wkich it follows that b and c are comparable. I f c <~ b <~ a, then b E X because X is convex, and if

b <~ c <~ a, then c E Y, a contradiction.

Fox lattices of the form Sub+(P) , not every ,ordering of free pairs produces a semilattice needed. In

order to be able to determine how ~ee pairs are to be ordered, we introduce the notion of a supersocle.

The idea is quite ~i~ple: to the zmiverse of the socle we add an element which is interpreted as the greatest

d e

f ~ gP3 0

Fig. 3

o / b / yl

Fig. 4

y,,

element adjoint to the desired lower semilattice P. Order free pairs in some way and then select those pairs

which belong to the "weak" order, i.e., are comparable in P. Correspondingly, redefine the relation -~ in

such a way as to leave pairs of comparable dements only, simultaneously expanding the domain of o. If the

order of free paizs is chosen properly, the reformed socle will satisfy both UDC and ZC.

We turn to the formal presentation. Let L be an atomistic lattice that satisfies (D2) and contains no

cycles. For ~" E {0, 1} and (a, b) E At(L) 2, we adopt the following notation:

(a,b) ifr=O; (a, b)" = (b ,a ) i f ," = ~.

For a socle 17 : (At(L); o, .-~), the basic relation t C_-,~ is defined in the following way. For any conjugate

elements a and b, a r b, we verify whether there exists an X-zigzag which begins with (a, b)" for some

sector X and some -r E {0, 1} and does not end in a free pair. If it exists and satisfies one of the conditions

(a) or (b), put (a, b)" e t. Introduce the set W = {(a,, b~) e A t ( L ) 2 / i = 1, k}, a list of all free paks with

ai # bi and {a,, bi} # {aj, b} , where i < k and i # j . For any j �9 2 k, the set W i - {(al, b i ) i ( i ) / i = 1, k}

is referred to as a j-scheme. It is easy to see that W i interprets a certain ordering of free pairs. For every

j E 2 k, complement the basic relation t by setting t i -- t U ((z, y) �9 A t ( L ) 2 / z E A, y �9 B, for some pair

(a, b) E Wi}. We call a j-scheme ~ru~hlike if, for any conjugate u, v �9 At(L), u ~. v, the set t~ contains only

one of the pairs (u, v), (v, u). The core of this section is the following:

Def in i t ion 2.5. Let L be an atomistic lattice which contains no cycles and satisfies (D2), let ~ -

{At(L); o,..~) be its socle, and let Wj, j �9 2 k, be a truthlike j-scheme. The supersocle ~ of the lattice L

wiih ~he scheme Wj is the system I? i = (At(L) U {~}; oj, ~i ) , ~ ~ At(L), of the same type as l?, in which

for any a, b, c E At(L) we have

O) (~,a), (a,,C) �9 " i ; (2) (a,b) �9162 a ,.. b and (a,b)" �9 t~ for some r �9 {0, I}; (3) ao j b = c if n o b = c;

(4) a 0 i b -- ~ if a ..- b and (a, b)" ~ q for any ~" �9 {0, I}. We give an example to illustrate the notion of a supersocle. Let Ps be the semilattice shown in Fig. 5a,

and L = Sub+ (Ps). Then the socle of L is the system ~5 = ({a, al, b, c, d, e}; o, -..), where

. . .= ((a,a~)", (a~,b)', (a,b)",(b,c)' , (:,d)", (a,d) ' ,

(a~, d)', (b, d)', (b, ~)', (a, ~)', (c, e)', (a~, e) T, (d, e)"/," �9 (0, 1}}

and

aoc-b, alOC- b, aloe--d.

b ~ d e a c

c e

Ps P~ 0 0

Fig. 5a Fig. 5b

It is easy to show that

t = {(al, b), (a, b), (c, b), (e, d), (al, d)}

is a basic relation, and W = {(a, al)}, where the pair (a, al) is B-free. Since

and

= (t u {(a, al)})' = t u d), e)

are antisymmetric, we have two trutMike schemes, Wo = {(al,a)~ and Wx = {(a, al)}. There are two

respective supersocles to consider:

~0 = (At(L) U {~}; o0, ~0/ and ~1 = (At(L) U {~); ol, ~1),

w h e r e ~ j = { ( z , y ) " / ( z , y ) E t j , T e { 0 , 1 } } , c o i e = c o i d = b o i e = b o id=~, j=O,l, aood=aooe=~. The system ~o coincides with the socle of the lattice L~ = Subv(P{), where the upper semilattice P{ is

obtained from P5 by adjoining the greatest element ~ (see Fig. 5a). Similarly, the semilattice P~ in Fig. 5b

corresponds to the system ~1.

Now we are in a position to formulate the main result of the paper.

T H E O R E M 2.6. A finite atomistic lattice L can be presented as the lattice Sub+(P) of partial upper

subsemilattices of a semilattice P iff L is biatomic, satisfies (D2), contains no cycles, and at least one of

the supersocles of the L satisfies UDC and ZC.

P r o o f . Necessity. Let L = Sub+(P). By Lemma 2.4, we can assume that < = <~. In accordance with

Proposition 1.4, the lattice L is biatomic, satisfies (D2), and contains no cycles. Thus we need only find a

supersocle satisfying UDC and ZC. Let Wi be the scheme which contains a free pair (a, b) iff a < b in P.

We show that the supersocle ~ = (At(L)U {~}; o~, ~i) with the scheme W~ satisfies the conditions required.

By Theorem 1.3, it suffices to show that ]P~ is isonmrphic to the socle of the lattice Subv(P~) for the (upper)

semilattice P~ -- P U {~} with the greatest element ~. Indeed, <~ = < by the definition of <~ and W~, from

which it follows that a ~i b iff a and b are comparable in P or ~ E {a, b}, i.e., if a and b are comparable in

P~. Suppose a and b are incomparable in P. If they have an upper bound in P, then a oi b - c for some

c E P. If o~ is a partial operation on the socle of the lattice Subv(P~), then clearly a o~ b - c. If a and b

do not have an upper bound in P, then a oi b = ~ by the definition of a supersocle. Since ~ is the greatest

element of P~, it is the unique upper bound for a and b in P~, i.e., a o~ b = ~. Similarly, if a o~ b = c, then

a oi b = c, as desired.

Sufficiency. Assume that some supersocle I~ -- (At(L) U {~}; oi, ~i) satisfies UDC and ZC. Consider

L 1 - S(]Fi), a lattice of subsystems of ~i. By Lemma 1.2, the system ]Pc is the socle of the atomistic lattice

L 1. Since L is biatomic and contains no cycles, L' inherits these properties. By Theorem 1.3, we then have

L' ~ Subv(S) for some upper semilattice S. Let S' -- SU{0} he a seml]attice with least element 0. Assume

that 0 is a nullazy operation on S', and Suhv(S') ~- Subv(S) is a lattice of subsemilattices of S' with 0. By

the definition of ]P~, the element ~ is comparable with all the elements from S t. If S t has one more element

W with this property, then (~, 7}) is completely free and, consequently, can be ordered arbitrarily, in which

case the lattice of upper subsemilattices remains unchanged (for details, see [9]). Without loss of generality,

we may assume that ~ is the greatest element in S'.

Consider the lower semilattice P = S' - {~}, all elements of which except 0 correspond to atoms in the

lattice L. We show that this correspondence induces an isomorphism L -~ Sub+(P). Since both lattices

are biatomic, contain no cycles, and satisfy (D2), by Lemma 1.2 we need only prove that their socles are

isomorphic.

Let IF = (At(L); OL, ~L) be the socle of L and S : (At(L); o,, ~ , ) be the socle of Sub+(P) : S u b + ( S ' -

{(}). Recall tha t ti denotes the basic relation on the socle F, expanded according to the i-scheme. Let

z ~L Y. If (z, y) E t~, then z ~ y, and hence the elements z and y are comparable in the semilattice S, and

none of the z, y is equal to ( by assumption. Then z and y are comparable in the semilattice S, whence

x ~ , y.

Now let (z, y) ~ t;. Then zo~y = ~ by definition. This means that the elements z and y are incomparable

in S, and their least upper bound is ~. Consequently, z and y do not have a common upper bound in

P = S - ~ } , which imphes z ~ , y. Conversely, let z ~ , y. If z and y are comparable in P = S - {~},

then they are also comparable in S, whence z ~ y, and so z "~L Y by definition. If z and y do not have

a common upper bound in P, then their least upper bound in S is ~. Consequently, z o~ y : ~, whence

z ~L Y by the definition of ~ .

Assume z o L y = z. Then z oi y = z, and consequently z = z + y in S. In addition, z r ~ by assumption,

so z = z + y in P = S ' - {~}, and hence z = zo, y. Similarly we can show that z % y = z implies z o L y = z.

In the remainder of the section, we show that in the class of finite lattices Sub+(P) , lattices of the form

Subv (P) are distinguished by a certain property shared by their atoms. For an atomistic lattice L which

contains no cycles and satisfies (D2), the set X C At(L) will be called decomposable if X is a disjoint union

of sectors Xi, i = 1, n, (we write X = O { X i / i --: 1, n}). It turns out that in lattices Subv (P) the set At(L)

is always decomposable . Actually, a more general s tatement is true.

L E M M A 2.7. Let L be an atomistic lattice which satisfies (D2) and contains no cycles. If the socle IF

of L satisfies UDC, then At(L) is decomposable.

P r o o f . Since L contains no cycles, At(L) is a union of limit sectors Xx . . . . ,Xn. Suppose tha t a E

Xi fl Xj , i # j. Then a # z i , z j and zi ~ zj (here, zi and z i are the bot toms of, respectively, X~

and Xi). Since a G X,, we have x, = ( ~ o ( t ~ _ l o ( . . . o ( t l o a ) . . . ) ) ) for s o m e t l , tk e At(L). Since tl t~

y ... z~ for all y 6 At(L), it follows that (a, z~) ./ ... 7 (z~, zi) is a left descent in IF. The left descent #1 #~.

(a, z i ) 7 . . . /~ (zj , z j) can be constructed in a similar way. Changing z i in the lat ter descent by z~ yields

a left descent again, which contradicts UDC.

I t is not hard to show that the condition of a set of a toms being decomposable is not only necessary

but also sufficient for presenting a lattice L as Subv(S), provided that L belongs to the class of lattices

Sub+(S).

10

P r o p o s i t i o n 2.8. Let L be a finite atomistic lattice. Then L ~- Subv(P) for a semilattice P if and

only if L -~ Sub+(S) for a lower semilattice S, and the set At(L) is decomposable.

P r o o f . By Lemma 2.7, we need only argue for the sufficiency. Let L ~ Sub+(S), At(L) = O(X/ i = 1,n}, and IF be the socle of L. It is obvious that a ~. b for any a E Xi, b E Xj , i ~ j . Let

P = 0p ~ X , ~ . . . ~ Xl be the ordinal sum of sectors treated as partially ordered sets. Obviously, P is

then a lower semilattice with the greatest element z l . By Lemma 1.2, Sub+(P) "~ Sub+(S). Besides, P is

an upper semilattice, and so Sub+ (P) = Subv (P) ~ L.

It should be noted that the condition of the set of atoms being decomposable makes consideration of

supersocles unnecessary.

L E M M A 2.9. If L is an atomistic lattice satisfying (D2) and if At(L) is decomposable, then the socle

of L and all of its supersocles simultaneously either satisfy or not UDC and ZC.

P r o o f . Suppose that the socle ]F of L satisfies UDC, and At(L) : (J{X.i / j : 1, s} for some sectors Xj,

j : 1, s. Consider a supersocle IFi : (At(L) U {~}; o~, "~i>. It is easy to show that a o~ b : ~ for any a E Xk tl t~ t .

and b E Xn~, k ~ m, (for details, see Lemma 3.2). Consider a descent (a, b) / (al , b ) / . . . ,,~ (a , , b) in ~i,

b ~ ~. Since a ~ i b, tk "~i b, k -- 1,n, and ~i C ..-, the given sequence is the beginning of a descent in ~, and t~+z

a, b, ~k E Xj for some j E ( 1 , . . . , s}. Suppose this descent can be continued in ]F, say, (a,,, b) / ( a ,+ l , b).

Then a,~+l = a,, o t , + l ( - - a,~ oi ~-+1), whence t,~+l E Xj. Since b .-~ t,,+l and the elements b and t ,+x

are conjugate by the sector Xj, we have b ~i t,~+l. Consequently, the same pair ( a ,+ l , b) continues also a

descent in IF~, a contradiction. Thus, if b ~ ~, then a descent in IFi is a descent in IF. This means that a last

pair of every such descent in ~i depends only on a pair with which it starts.

We turn to the case b - - ~ . I f a E Xi, t l E Xk, and k ~ j , t h e n a l = ~ , and the descent consists of

just two pairs. If a, ~1 G Xj, then al E Xj, since otherwise we arrive at a contradiction with the condition

that At(L) is decomposable. Thus, either t , , a,, E Xj, or t , E Xk, k r j , and a~ = ~. In the former case

At(L) : Xj, and so descents in IF~ beginning with the pair (a, ~) will always end in the pair (z j , ~). In the

latter case such descents always end in (~,~). Indeed, if (a, ~ ) / . . . / (c, ~) is the beginning of a descent

in IFi, and c ~ ~, then c, a E Xj by the above argument. Since, in addition, there exists a sector X~, k ~ j ,

the end of the descent is (c,~) 7 (~,~)-

Let ~ satisfy ZC and let (a, b ) , . . . , (c, c) be an exact zigzag in IFi. If a, b r ~, then c r ~, as was shown

above. This means that zigzags in ~i beginning with (a, b) are also zigzags in l?, and so ZC is satisfied in

l~ . If a = ~ or b -- ~, then the pair (a, b) cannot b e the beginning of both the left and right zigzags, and so

ZC is satisfied in this case as well.

We will prove the converse statement which says that if UDC and ZC are satisfied in some supersocle

of IF~, they are also satisfied in ~. Let a ~ b and (a, b ) / 2 . . . / ~ (a~, b) be a descent in ~. First, consider the

case where a and b are conjugate by the sector X i. Since c ,.~ d for any nonconjugate elements c, d G At(L),

a l - a o t~ implies t~ E X i . Ifa~ ~ X~, k r j , then a G X i AXe, which contradicts the decomposability of

At(L). So a I G X] and, similarly, ~ 2 , a 2 , . . . , t , a , ~ X i. Since b " ~,n, rn : 1,n, and the elements b and tra

are conjugate by the sector Xi, b "~i tm in the supersocle of IFi. And we can take (a, b),..., (a~, b) to be the t~+x

beginning of some descent in IF~. Assume that this descent can be continued in IF~, say, (a~, b) / (a~+~, b).

Then t,,+x "~i b, whence ~,,+x G Xi. If a,+x = a~ oi t,,+x ~ ~, then a,+x = a~ o t ,+x, and so the pair

(a~+x, b) continues a descent in IF. If a~+x = ~, then a , and t ,+x cannot be conjugate, a contradiction.

Thus, in the case where a and b are conjugate any descent in IF beginning with (a, b) is a descent in F~.

11

Now let a E Xj and b E Xk, j r k. Then a descent in IF ends in the pair (zj, b), i.e., UDC is satisfied in

this case as well.

Finally, consider an exact zigzag (a, b),..., (c, c) in liD. We see that a and b are conjugate by the sector

C, and so this sequence is a zigzag in ~i. Consequently, ZC is true for ]?.

L e m m a 2.9 and Proposi t ion 2.8-allow us to assert t ha t Theorem 1.3 is a particular case of Theorem ~.6

~ailored for atomis~ic la~ices L in which the se~ At(L) is decomposable.

S T R U C T U R E O F A S S O C I A T E S E M I L A T T I C E S

Semilat t ices are called associate if their congruence latt ices are isomorphic. In the present section, we

give a descript ion of finite associate semilattices. For the semilatt ices /P ; -< /wi th _< = _<~, this has a l ready

been done in T h e o r e m 2.6. Semilatt ices with this p rope r ty will be called broad. I t follows f rom T h e o r e m

2.6 that if L ~ Sub+(P) [L ~ ~ Con(P), where L ~ is a lattice dually isomorphic to L], then all broad

semilattices P' with Sub+ (P') -~ L [Con(P') ~ L ~] are induced by some orderings of elements of free pairs.

We know that suitable orderings are those for which the corresponding supersocles ]Fi satisfy UDC and ZC.

Verification of these properties furnishes an algorithm which determines all broad semilattices P', given

the congruence lattice L ~. The description of broad associate semilattices is presented in the proposition

below, which is a consequence of Theorem 2.6.

P r o p o s i t i o n 3.1. Let P1 = (P; _<x) and P2 -- <P; <_2) be broad semilat t ices wi th the schemes Wt and

W2, respectively. (V~i = {(a, b) �9 p 2 / a -<i b and the pair (a, b) is free), i = 1, 2). I f Con(P1) -~ Con(P2) ,

then P2 may be obtained from Pt by replacing the scheme, i.e., <2 = (((<-'t)c - (WI)) U (W2))*, where

(wi) = y) �9 p2/ �9 a, y �9 B for some pai (a, b) �9 i = 1, 2.

For associate semilattices to be described completely, wc can apply Lemma 2.4 which states that, for

any semila t t ice P = (P; _ ) , there exists a uniquely de termined associate broad semila t t ice P* = (P; -<*).

In this case P is said to be reduced to P*. Our current goal is to describe all semilat t ices reduced to a

given b road semilat t ice P* . The groundwork for the p roof is L e m m a 3.2 which sheds new light on how <__~

is s t ruc tured .

We in t roduce some new notions. A nonvoid subset of an atomist ic latt ice is called sec~oral if it is a

union of one or several l imit sectors. A sectoral subset is said to be minimal if it is not a disjoint union

of other sectoral subsets. Clearly, if L has no cycles, then At(L) can be uniquely represented as a disjoint

union of min imal sectoral subsets. If U C P and p C_ p2 , we denote p f3 U 2 by p[u.

L E M M A 3.2. Let L = Sub+(P, -<) and At(L) = O{U~/ i = 1, s ) , where U~ are min imal sectoral

subsets. T h e n _<~ = U{_<Iu , / i = 1, s} U {(0v, z) / z �9 P}.

P r o o f . We show tha t _C. If a _< b and a and b are conjugate, then, obviously, a, b E Ui for some i < s.

Moreover, let b and c be conjugate by a sector S with ~ (a, S). Then, as shown in L e m m a 2.1, a _< e, which

in view of S C_ Ui yields (a ,c) E <[u, , and we are done.

Conversely, assume tha t (a, b) �9 _<Iv, for some i <_ s. By definition, U~ = U { X v / p = 1, k} for some limit

sectors X~,. Since Ui is minimal , one can find a sequence b = to, t l , . . - , t,~ = a of e lements f rom Ui in which

any neighbors Q and *i+t are conjugate, i.e., belong to the same sector Xp~. Let i = rn be the min ima l

index for which a �9 Xv,. If m = 0, then a and b are conjugate by the sector Xpo , and hence (a, b) �9 -<;.

If m > 0, then a ~ Xpo,...,Xp,~_ ~. Moreover, since a -< b, by Lemma 2.1 we have a -< Q,...,a -< tin,

and also t , , �9 Xp. . , i.e., a and t are conjugate. Therefore, (a, b) is ob ta ined by the fan shift f rom (a, t,n),

whence (a, b) �9 _<~.

12

It follows from Lemmas 3.2 and 2.1 that all the semilattices reduced to a given broad semilattice

P* = (P; <~) are composed of the same "blocks" UI, ...,U,, which are the minimal sectoral subsets on

which the partial orders _~I - ~lu~,..-,-<* = ~ [u. are defined properly. We need to find all possible

configurations which these subsets form in a semilattice. Notice that if z E Ui, y E U i, and i ~ j, then z

and y should either be comparable or have no common upper bound in any semilattice P reduced to P*.

We call Ui and Uj incomparable in P if so are z and y for any z E Ui and y E Uj. It is clear that P* is

exactly that semilattice in which Ui and Uj are incomparable for any i # j . All other configurations formed

in a semilattice P by Ui and Uj with i # j are described as follows. We say that trj is grafted to Ui at point

t E U~ if t <__ z for all z G Uj, and t ' and z are incomparable for all (t', t) ~t <~ and all t ' E U~ with z E Uj.

In Fig. 6, for example, the set U2 = {t~2, a, b} is grafted to U1 = {ttl, c, t} at point 4.

L E M M A 3.3. Let P = (P; <) be the semilattice reduced to a broad semilattice P* = (P; <~) and

let P = 0{U~/ i = 1, s} be the decomposition into minimal sectoral subsets. Then for any U~ and Uj,

i # j < s, either Ui and Uj are incomparable in P, or Uj is grafted to U, at some point t E U~, or Ui is

grafted to U i at some point y E Uj.

P r o o f . Suppose that Ui and Uj are not incomparable in P. To be specific, put z < y for some z E Ui

and y E Ui. We can assume z to be the greatest element in Ui satisfying z < y. We show that z < t for

all t E Uj. Since Uj is minimal, for any t, there exists a sequence vo = y, v l , . . . , v~ = t of elements in

Uj such that any neighbors vp and vp+~ are conjugate by some sector Xjp C_ Uj. Since z ~t Uj, we obtain

z < ~/= vo, z < v l , . . . , z <_ vk = t, as desired. Now let ( z ' , z ) ~ <~, z ' E Ui, and y' E U;. If z ' < y', then

z' < t for all t E Uj by the preceding argument. In particular, z ' < y, which contradicts the choice of z. If

y' < z ' , then y' < t for all t E Ui, which contradicts z < y'. Therefore, z ' and y~ are incomparable in P.

L E M M A 3.4. ( a ) I f Ui is grafted to Uj at the point y E Uj and Uj is grafted to U~ at the point s 6 U~,

then Ui is grafted to U~ at point s.

(b) If U~ is grafted to U~ and to U~, then either Ut is grafted to U~, or Uj is grafted to U~.

P r oo f . (a) Clearly, s < t for all t 6 Ui. We must show that s' and t are incomparable whenever

(s', s) ~ <~, s' E U~, and t ~ U~. By hypothesis, s' and y~ ar e incomparable whenever (s', s) r <~, y' ~ Uj,

and y < t for all t 6 Ui. I f s ' < t for some s' 6 U~, t E Ui, and (s',s) q~ <~, then t will be a common upper

bound for incomparable elements s' and !/, a contradiction with s' ~ y. Evidently, t < s ~ also fails, since

otherwise we would have t < s.

(b) Let Ui be grafted to U~ at point y and to U~ at point s. Then y and s have a common upper bound,

and the required statement will then follow from Lemma 3.3.

By Lemmas 3.3 and 3.4, on the set {0, 1, 2 , . . . , s} we can define a strict partial order -< by putting 0 -< i

~2

0

Fig. 6

13

for any i r 0, i <_ s, and j -~ i whenever Ui is grafted to Uj at some point y E U i. Then the partially

ordered set i2 = ({0, 1 , . . . , s}); -~) is a tree by (b) of Lemma 3.4, and y is a common point at which Ut is

grafted to Us for all k with i -< k -< j or j -~ k, by (a) in Lemma 3.3.

With this observation in mind, we show how to construct a semilattice reduced to a broad semilattice,

given the set U : { U x , . . . , U,} of its minimal sectoral subsets. Let f / : ({0, 1 , . . . , s}; -~) be some tree with least element 0, i.e., it is a lower semilattice in which two elements have a common upper bound iff they

are incomparable. Assume that I = {( i , j ) E f / 2 / j covers i}. To each pair (i,j) E I, assign an arbi t rary

element Ysj E Us and put Y : {y~j/(i, j) E I}. Given the triple (f~, U, Y), we construct a semi]attice (P; ___),

denoted hereinafter by ~3(fl, U, Y) and referred to as a ~ree with branches U1, . . . , U,. The universe of this

semilattice is the set P = {0p} U {Us~ i = 1, s}; ~[u~ coincides with the order ~s on the corresponding

sectoral subset. Define Ui and U 1 to be incomparable if so are i and j in fl and assume Uk to be grafted to

Us at point y iff y -- Ys, for some s -~ k.

P r o p o s i t i o n 3.5. Let P* = (P; _<*) be a broad semilattice with the set U of minimal sectoral subsets

U1, . . . , Us. Then, for any semilattice P reduced to P*, there exist an (s § 1)-element tree f~ and a subset

Y C P such that P = ~(12, U, Y).

The p r o o f follows from Lemmas 3.2-3.4.

Finally, combining Propositions 3.1 and 3.5, we obtain the following:

T H E O R E M 3.6: Let P~ = (P; <~) and P2 -" (P;_<2) be the lower semilattices with the schemes

W1 and W2, respectively. The lattices Con(P~) and Con(P2) are isomorphic iff/>1 is a tree with branches

U1, . . . , U~, P2 is a tree with branches V1, . . . , V~, and for any i ~ s, the sectoral subset V~ is obtained from

the corresponding Us by replacing the scheme, i.e., <_2[v, = (((_<l]u,)c - (W1)[u,) (J (W2)lv~)*.

F I N A L R E M A R K S

We finish with some remarks concerning the problem of algorithmic complexity. Let L be a finite

atomistic lattice with n elements and k atoms (k < n <_ 2k). Which time is necessary to verify whether a

given lattice can be presented in the form Con~ for some semilattice P? The algorithm suggested in [3]

requires that we examine all subsemilattices of L, in which case the time consumed depends exponentially

on n. If, however, we apply the algorithm presented in this paper, the time needed to determine whether a

lattice is biatomic, has no cycles, and satisfies (D2) as well as UDC and ZC varies polynomially in n. The

number of supersocles themselves depends on k exponentially, and so direct application of the algorithm

from Theorem 2.6 needs time that varies ezponentially with k. We can show, however, that the number of

supersocles to be examined to produce a negative answer is actually not more than n. Thus, an improved version of our algorithm will require ~ime tha~ is polynomial in n. Also we have quite a number of tricks

at hand to make this algorithm less time-consuming. For instance, we can expand a basic relation and

arrange examination of/-schemes in such a manner that a considerable number of cases are not searched at

all. These modifications, accelerating the work in particular cases, will not alter the polynomial character

of the algorithm.

R E F E R E N C E S

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(1973).

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Saratov Univ., Sa~atov (1971), pp. 11-21.

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