Simultaneous congruence representations: a special case

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1 Algebra univers. 54 (2005) /05/ DOI /s c Birkhäuser Verlag, Basel, 2005 Algebra Universalis Mailbox Simultaneous congruence representations: a special case William A. Lampe Abstract. We study the problem of representing a pair of algebraic lattices, L 1 and L 0, as Con(A 1 ) and Con(A 0 ), respectively, with A 1 an algebra and A 0 a subalgebra of A 1, and we provide such a representation in a special case. 1. Introduction Back in 1971, Ervin Fried posed the problem of representing a pair of algebraic lattices, L 1 and L 0,asCon(A 1 ) and Con(A 0 ), respectively, with A 1 an algebra and A 0 a subalgebra of A 1. This seems to be a very hard problem in general. Note that if Con(A 1 ) has only one element, then Con(A 0 ) has only one element. It is known that if L is an algebraic lattice having a compact 1, then L is isomorphic to the congruence lattice of a groupoid A = A,, where is a binary operation. We settle Fried s question positively in the case that both L 1 and L 0 have compact 1 s. Theorem 1. Suppose L 1 and L 0 are algebraic lattices and L 1 has at least 2 elements and L 1 and L 0 both have compact 1 s. Then there is a groupoid A 1 = A 1, and a subgroupoid A 0, so that L i is isomorphic to Con(A i ),fori =0, 1. This solves the simultaneous representation problem for pairs of finite lattices, as we point out in the Corollary 2. Suppose L 1 and L 0 are finite lattices and L 1 has at least 2 elements. Then there is a groupoid A 1 = A 1, and a subgroupoid A 0, so that L i is isomorphic to Con(A i ),fori =0, 1. Recall that a semilattice homomorphism is 0-separating iff it is 0-preserving and sends nonzero elements to nonzero elements. Presented by B. Jónsson. Received September 11, 2004; accepted in final form January 7, Mathematics Subject Classification: 06B15, 08A30. Key words and phrases: congruence lattice representations. 249

2 250 W. A. Lampe Algebra univers. We suppose A 1 is an algebra and A 0 is a subalgebra of A 1. Let S be a set of pairs. To simplify notation we will let [S] Ai denote the congruence relation of A i generated by S. Let Θ be a congruence relation of A 0. Then the map which sends Θ [Θ] A1 is a complete, 0-separating join homomorphism from Con(A 0 )intocon(a 1 ) sending compact elements to compact elements. Such a homomorphism is determined by its action on the semilattice of compact elements. Since the only subdirectly irreducible semilattice is the two element one, there is always at least one 0-separating homomorphism from any given semilattice to any other as long as the latter has at least two elements. More precisely, the theorem we will prove is the following. Theorem 3. Suppose L 1 and L 0 are algebraic lattices and L 1 has at least 2 elements and L 1 and L 0 both have compact 1 s. Suppose also that γ is a complete, 0-separating, 1-preserving join homomorphism from L 0 to L 1 sending compact elements to compact elements. Then there exists a groupoid A 1 = A 1, and a subgroupoid A 0 and an isomorphism σ i from L i onto Con(A i ) such that [σ 0 (x)] A1 = σ 1 (γ(x)) for any x L 0. Theorem 1 follows from Theorem 3 as long as L 0 also has at least 2 elements, because in that case there is always at least one such homomorphism γ, namely the one that sends 0 to 0 and everything else to 1. In case L 0 = 1, Theorem 1 just asserts that there is a pointed groupoid whose congruence lattice is isomorphic to L 1, and that is a corollary of Theorem 2 of [3]. Ervin Fried had suggested that one could settle the general problem by first building a representation of L 0, and then somehow building a representation of L 1 on top of that. In general, that seems difficult, but it is the approach we take in proving Theorem Preliminaries Suppose A is a partial algebra with a distinguished element 0, and suppose that H is a nonempty set of congruences of A with the property that the H-closure of S =[S] H =Θ H (S) exists for each finite subset S of A A. We will say compact strongly equals principal in H iff the following two properties are satisfied: (1) if S is a finite subset of A A, then there are a, b A so that Θ H (a, b) = Θ H (S); (2) if S and T are finite subsets of A A, then there are a, b A so that Θ H (a, 0) = Θ H (S), Θ H (b, 0) = Θ H (T )andθ H (a, b) =Θ H (S) Θ H (T ) (= Θ H (S T )).

3 Vol. 54, 2005 Special simultaneous representations 251 Suppose C is a set of equivalence relations on A and the C closure of each element of A A exists. If D A, thenwesay x is the closest thing to y in D, modulo C andwewrite iff the following hold: x CLS y (in D, mod C) (i) x D; (ii) Θ C (x, y) Θ C (z,y) for every z D; (iii) x = y if y D. We say D is a C-closed subset of A (or a closed subset of A) iffeitherd A and D = or for every a A there is a c satisfying c CLS a (in D, mod C). A is a partial pointed groupoid iff A is a partial groupoid having an idempotent element 0. What follows, (#), is a list of assumptions we will need to make. (#) (A) A is a partial pointed groupoid. (B) H Con(A), and A H. (C) H is an algebraic closure system. (D) There is a D A such that D D =Dmn(, A). (E) D is H-closed. (F) For every a, b A there are closest elements c, d in D satisfying Θ H (a, b) Θ H (c, d). (G) For every x, y, u, v D either Θ H (ux, vy) = Θ H (u, v) Θ H (x, y) orelseθ H (x, y) =A A. (H) Compact strongly equals principal in H. The next lemma is a special case of Lemma 4 of section 3 of [3]. Lemma 4. If L is an algebraic lattice with compact 1, then there are A and H satisfying (#) with L isomorphic to H;. The next theorem is implicit in the proof of Theorem 2 of [3]. Theorem 5. If A and H satisfy (#), then there is a pointed groupoid A such that: (i) A is a partial subgroupoid of A ; (ii) Θ [Θ] Con(A ) is an isomorphism from H; onto Con(A); (iii) Con(A ) and A satisfy (#).

4 252 W. A. Lampe Algebra univers. 3. The main proof Suppose L 1 and L 0 are algebraic lattices, L 1 has at least 2 elements and L 1 and L 0 both have compact 1 s. Suppose also that γ is a complete, 0-separating, 1- preserving join homomorphism from L 0 to L 1 sending compact elements to compact elements. (So L 0 also has at least two elements.) By Lemma 4 and Theorem 5, there is a groupoid A 0 such that A 0 and Con(A 0 ) satisfy (#) and such that Con(A 0 ) is isomorphic to L 0 under an isomorphism σ 0 from L 0 onto Con(A 0 ). Suppose L is any algebraic lattice. Cmp(L) denotes the set or semilattice of compact elements of L. Let C denote the set of nonzero, compact elements of the algebraic lattice L 1, and set B 0 = A 0 C. (Here we assume A 0 and C are disjoint.) We let B 0 = B 0, be the partial groupoid with being the same exact function in B 0 as in A 0. Hence, Dmn(, B 0 )=A 0 A 0. C {0 L1 } =Cmp(L 1 ) is a join semilattice. Let I be an ideal of this semilattice, and set Ψ I = (σ 0 (x) :x Cmp(L 0 )andγ(x) I). Ψ I is a congruence of A 0. From here on we identify 0 L1, the zero of L 1, with 0, the distinguished, idempotent element of A 0.Nowweset Φ I =Ψ I ((0/Ψ I ) I) 2 C where C is the equality relation on C and (0/Ψ I ) is the congruence class of 0 under the relation Ψ I.WesetH 0 = {Φ I : I is an ideal of Cmp(L 1 )}. Lemma 6. Under the above assumptions, the following hold. (i) There is an isomorphism τ 1 from L 1 onto H 0,. (ii) For any x L 0, [σ 0 (x)] H0 = τ 1 (γ(x)). (iii) B 0 and H 0 satisfy (#). Proof. The map sending I Φ I is obviously order preserving. Note that Φ I (C {0}) 2 = I 2 C. It follows that the map I Φ I is an order isomorphism. Let x L 1, and set I x = {c Cmp(L 1 ) : c x}. Asiswellknown,the map sending x I x is an isomorphism from L 1 onto the lattice of ideals of the semilattice Cmp(L 1 ). Set τ 1 (x) =Φ Ix.Thenτ 1 is the composition of two isomorphisms and is thus an isomorphism, so (i) holds.

5 Vol. 54, 2005 Special simultaneous representations 253 Next, we Claim. The following hold. (1) If (I j : j J) is any family of ideals of the semilattice Cmp(L 1 ), then (ΨIj : j J) =Ψ T (I j:j J). (2) If (I j : j J) is any family of ideals of the semilattice Cmp(L 1 ), then (ΦIj : j J) =Φ T (I j :j J). (3) If (Φ Ij : j J) is an up directed family, then (ΦIj : j J) =Φ S (I j :j J). (4) H 0 = {Φ I : I is an ideal of Cmp(L 1 )} is an algebraic closure system in which compact strongly equals principal. Suppose (I j : j J) is a family of ideals of the semilattice Cmp(L 1 ). The map sending I Ψ I is obviously order preserving. So (Ψ Ij : j J) Ψ T (I j:j J). So we let a, b (Ψ Ij : j J). So the compact congruence Θ(a, b) (Ψ Ij : j J). So Θ(a, b) Ψ Ij for each j J. From the definition of the Ψ I s and the compactness of Θ(a, b) weseeforeachj J that there are x j,1,...,x j,k Cmp(L 0 )with Θ(a, b) σ 0 (x j,1 ) σ 0 (x j,k )andwithγ(x j,i ) I j. For some y, Θ(a, b) = σ 0 (y). We have σ 0 (y) =Θ(a, b) σ 0 (x j,1 ) σ 0 (x j,k )=σ 0 (x j,1 x j,k ). Since σ 0 is an isomorphism, we conclude that y x j,1 x j,k. and thus we have γ(y) γ(x j,1 x j,k )=γ(x j,1 ) γ(x j,k ). But each γ(x j,i ) I j and I j is an ideal, and so γ(y) I j,foreachj J. Therefore γ(y) (I j : j J). Then a, b Θ(a, b) =σ 0 (y) Ψ T (I j:j J), finishing the proof of (1). Suppose (I j : j J) is a family of ideals of the semilattice Cmp(L 1 ). The map sending I Φ I is order preserving. So (Φ Ij : j J) Φ T (I j :j J). So we let a, b (Φ Ij : j J). So a, b Φ Ij for each j J. There are three possibilities. First is the possibility that a, b (C {0}). So a, b (Ij 2 C)foreach j J. If a, b / C,then a, b Ij 2 for each j J, andso a, b ( (Ij 2 : j J)). In either case, a, b Φ T (I j:j J). Second is the possibility that a, b A 0.Then a, b Ψ Ij for each j J. By(1) we have a, b Ψ T (I j:j J) Φ T (I j:j J). Third is the possibility that a A 0 and b C or vice versa. So we must have a, b ((0/Ψ Ij ) I j ) 2 for each j J. So a (0/Ψ Ij ) I j,foreachj J. But a A 0,ora C, but not both. So a (0/Ψ Ij )foreachj J, ora I j for

6 254 W. A. Lampe Algebra univers. each j J. In either case, a (( (0/Ψ Ij : j J)) ( (I J : j J))). Moreover (0/ΨIj : j J) =0/( (Ψ Ij : j J)). So a (0/( (Ψ Ij : j J)) ( (I J : j J))). Similarly, b (0/( (Ψ Ij : j J)) ( (I J : j J))). So we have a, b (0/( (Ψ Ij : j J)) ( (I J : j J))) 2. From (1) we have (Ψ Ij : j J) =Ψ T (I j:j J). Andso a, b (0/Ψ T (I j:j J) ( (I J : j J))) 2 Φ T (I j:j J). This finishes the proof of (2). Suppose (Φ Ij : j J) is an up directed family. Then (Φ Ij (C {0}) 2 : j J) is an up directed family. But Φ Ij (C {0}) 2 = Ij 2 C.So(I j : j J) isanup directed family of ideals of the semilattice Cmp(L 1 ). So (I j : j J) isanideal of the semilattice Cmp(L 1 ). Since the map sending I Φ I is order preserving, (ΦIj : j J) Φ S (I j:j J). The reverse inequality is easy to prove and left to the reader. So (3) is true. By (2) and (3) we have that H 0 = {Φ I : I is an ideal of Cmp(L 1 )} is an algebraic closure system. Let S and T be finite subsets of B 0 B 0. Now Θ H0 (S) is a compact member of H 0. So there is a c Cmp(L 1 )sothatθ H0 (S) =Φ (c], where (c] is the ideal generated by c. Now Θ H0 (S) =Φ (c] =Θ H0 (c, 0). Similarly, Θ H0 (T )=Θ H0 (d, 0) for some compact d Cmp(L 1 ). Now Θ H0 (S T )= Θ H0 (c, 0) Θ H0 (d, 0) = Φ (c] Φ (d] =Φ (c d] =Θ H0 (c, d). So compact strongly equals principal in H 0, and so (4) is satisfied. Let x L 0. Clearly, [σ 0 (x)] H0 =Φ I,whereI is the smallest ideal such that σ 0 (x) Ψ I = (σ 0 (y) :y Cmp(L 0 )andγ(y) I). It is easy to see that I is the ideal of Cmp(L 1 ) generated by (γ(z) :z Cmp(L 0 )andz x). Moreover, this ideal is I γ(x) = {d Cmp(L 1 ):d γ(x)}. So[σ 0 (x)] H0 =Φ Iγ(x) = τ 1 (γ(x)), which establishes (ii) of the Lemma. Now we turn to showing that (#) holds for B 0 and H 0. By construction, B 0 is a partial pointed groupoid, and (A) holds. (4) of the claim establishes that (C) and (H) hold. Dmn(, B 0 )=A 0 A 0, and so (D) holds. Now each Φ I Con(B 0 )sinceφ I (A 0 A 0 )=Ψ I Con(A 0 ) and since A 0 is a subgroupoid of B 0.Also,Φ {0} = B0.So(B)holds. If a A 0 and b (B 0 A 0 ), then a 0 b (mod Θ H0 (a, b)). So A 0 is H 0 -closed, and (E) holds, and (F) follows easily. Suppose x, y, u, v A 0 and either u, v / Θ H0 (ux, vy) or x, y / Θ H0 (ux, vy). Then in A 0,either u, v / Θ(ux, vy) or x, y / Θ(ux, vy). Since A 0 and Con(A 0 ) satisfy (#), then Θ(x, y) =A 0 A 0.Now,Θ H0 (x, y) =[A 0 A 0 ] H0 =[σ 0 (1)] H0 = τ 1 (γ(1)) = τ 1 (1) = B 0 B 0, by (i) and since γ(1) = 1 by hypothesis. Thus(G) holds, ending the proof of the lemma. Theorem 3 now follows from the construction, Lemma 6 and Theorem 5.

7 Vol. 54, 2005 Special simultaneous representations Concluding remarks A pinched lattice is an algebraic lattice having a set I of compact elements such that I = 1 and such that each compact element of L is comparable with every element of I. An algebraic lattice with a compact 1 is pinched, and so is any algebraic chain. Theorem 3 can be generalized easily to pinched lattices, in part because each pinched lattice is the congruence lattice of a pointed groupoid. Theorem 7. Suppose L 1 and L 0 are pinched algebraic lattices and L 1 has at least 2 elements. Suppose I i is a set of nonzero, compact elements of L i with I i =1 and with each compact element of L i comparable with every element of I i.suppose also that γ is a complete, 0-separating join homomorphism from L 0 to L 1 sending compact elements to compact elements and sending I 0 into I 1. Then there exists a groupoid A 1 = A 1, and a subgroupoid A 0 and an isomorphism σ i from L i onto Con(A i ) such that [σ 0 (x)] A1 = σ 1 (γ(x)) for any x L 0. Reference [3] contains other theorems representing various distributive lattices as congruence lattices of pointed groupoids. Analogues of Theorem 7 are provable for these cases as well. The general simultaneous representation problem remains open. It is not even known if every algebraic lattice is the congruence lattice of an algebra having a one element subalgebra, Moreover, one cannot solve the general problem without implicitly solving the latter problem. References 1. G. Grätzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math (Szeged) 24 (1963), W. A. Lampe, The independence of certain related structures of a universal algebra. I., Algebra Universalis 2 (1972), , Congruence lattices of algebras of fixed similarity type, II, PacificJ.Math.103 (1982), B. Šešelja and A. Tepavčević, Weak congruences in universal algebra, Institute of Mathematics Novi Sad, William A. Lampe Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822, U.S.A. bill@math.hawaii.edu