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Notes on Related Structures of a Universal Algebra

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Notes on Related Structures of a Universal Algebra Pacific Journal of Mathematics NOTES ON RELATED STRUCTURES OF A UNIVERSAL ALGEBRA WILLIAM A. LAMPE Vol. 43, No. 1 March 1972 PACIFIC JOURNAL OF MATHEMATICS Vol. 43, No. 1, 1972 NOTES ON RELATED STRUCTURES OF A UNIVERSAL ALGEBRA WILLIAM A. LAMPE The related structures of a universal algebra % that are studied here are the subalgebra lattice of % the congruence lattice of %, the automorphism group of % and the endo- morphism semigroup of 51. Characterizations of these struc- tures known, and E. T. Schmidt proved the independence of the automorphism group and the subalgebra lattice. It has been conjectured that the first three of the structures listed above are independent, i.e., that the congruence lattice, subalgebra lattice, and automorphism group are independent. One result in this paper is a proof of a special case of this conjecture. Various observations concerning the relationship between the endomorphism semigroup and the congruence lattice are also in this paper. In the last section a problem of G. Gratzer is solved, namely that of characterizing the endomorphism semigroups of simple unary algebras. (An algebra is simple when the only congruences are the trivial ones.) The characterizations of the various related structures are as follows: the congruence lattice is an arbitrary algebraic lattice the subalgebra lattice is an arbitrary algebraic lattice; the auto- morphism group is an arbitrary group; the endomorphism semigroup is an arbitrary semigroup with identity. The "independence of the automorphism group and the subalgebra lattice" is more precisely phrased as: for each pair <©, 8>, where © is a group and 8 is an algebraic lattice with more than one element, there is an algebra 2t with © isomorphic to the automorphism group of 21 and with S isomorphic to the subalgebra lattice of the same algebra St. All statements about the independence of related structures will be phrased in this way. Mentioned above was a proof of a special case of the independence of the triple consisting of the automorphism group, the subalgebra lattice, and the congruence lattice. As a corollary one gets a proof of a special case of the independence of the pair consisting of the auto- morphism group and the congruence lattice. E. T Schmidt published what was supposed to be a proof of the independence of this pair of structures. But, his proof [10] was incorrect. (See e.g. Exercise 31 of chapter 2 of [2]) The author has just completed a proof of the independence of this pair [8]. The terminology essentially conforms to that in [2]. ω(oτ ωA) will denote the equality relation on the set A, and φr cA) will denote 189 190 WILLIAM A. LAMPE the total relation. Θ(a0, αj will represent the smallest congrucence collapsing α0 and aλ. 2 = <L, Λ, V> will denote a lattice. (£(21) = <^(2t); S> will denote the congruence lattice of 2t @(Sί) = <^(2t); e> will denote the subalgebra lattice of St. ©(21) = <G(2t); °> will denote the automorphism group of St. @(2t) = <152ί); o> will denote the endo- morphism semigroup of 21. An important algebra for dealing with endomorphism semigroup and automorphism group problems is the algebra of left multiplications S(@) of the semigroup @. The operations are all left multiplication maps and the endomorphisms are all right multiplication maps. As in Cayley's Theorem, the semigroup of right multiplications of @ is isomorphic to @. Many of the details of the proofs which are left out can be found in the author's dissertation [6J. The various characterizations men- tioned above can be found in [1], [2], [3]. E. T. Schmidt's result on the independence of the automorphism group and subalgebra lattice is found in [11]. 1* The property restricting the representation of <©, 80, Sx> as <©(St), Let 21 = (A; F} be an algebra. The lattice 8 is assumed to be an algebraic lattice. Let αeL, and let (x^iel) be a family of elements of L. Essentially the property mentioned in the heading is: there exist α0, ax € A such that for any x Φ a0 and for any congruence β, if aQ = x(θ)y then a = a^θ). We will give a generalization of this property and a property of the congruence lattice equivalent to the more general property. Also, the class of algebraic lattices having the equivalent property will be discussed. Let α0, ate A with a0 Φ αx. (**) There exists a partition {Ao, AJ of A such that a^eA^ and for any (x, y} e AoxAl9 θ(aθ9 a,) ^ θ(a?, y). ( * ) If α ^ V (χi\i e I)> then a <£ α?< for some i. Notice that the originally stated condition is a special case of (**) where AQ — {α0}. Obviously, if an element a of 8 has property (*), then a is complete-join irreducible. Also, a has property (*) if and only if α's dual ideal is completely prime. PROPOSITION 1. // property (**) is satisfied for <α0, αx>, β(α0, αx) satisfies property (*) m £Λβ congruence lattice of 2t. RELATED STRUCTURES OF A UNIVERSAL ALGEBRA 191 REMARK. This statement was first observed by G. Gratzer. Proof. Suppose that (Φi\iel) is a family of congruences and that θ(a0, di) £ V (Φ<|ί ei). There exists a sequence a0 — zQ, •••, znf = aly with Zj e A such that zs = zj+ι{Φi3) for some iά e I. Since α0 e Aoy αx e Au and {Ao, AJ is a partition of A, there is a k such that s* e Ao and sA+1 G Alβ So <9(α0, αj £ θ(st, zk+1) £ Φ4jfc. PROPOSITION 2. // there is a congruence θ different from co having property (*), then θ — θ(a0, αj for some a0, ax in A with a0 Φ at and property (**) is satisfied for <α0, α^. Proof. Always θ = \/(θ(x, y)\x = y(θ)). Since θ has property (*), θ = θ(aj, y) for some x,yeA. Fix α0, αx such that θ = θ(α0, αj. Set 50-: {x\Θ(x,a0) g ft - {y\θ(y,ad g θ(α0, αx) ft= Set Ao = Bo and Λ = ft U ft. It follows that 40 Π 4i = 0. Clearly, aQ e Ao and α: € A:. Also A = A1 U A2. Let xoeAo and ^eij and consider θ(α?0, α^). First suppose that α?! G ft. Thus, θ(ίc0, α0) ^ θ(α0, αj and Θfe, α:) g Θ(α0, α^. Now, since θ(aQ, αx) £ θ(ί»o, α0) V θ(αj0, a?i) V <9(^i, αx) and since β(α0, αt) has (*), we have that β(α0, αx) £ θ(aj0, »i) Now suppose that ajj e ft. So @(α0, αx) £ Θ(a0, xλ) £ <9(α0, x0) V ©(OJO, «I), and thus, Θ(a0, a,) £ ®(a?0, »i) Combining these two propositions with the congruence lattice characterization theorem, we get the following statement. PROPOSITION 3. If 2 is an algebraic lattice, the following are equivalent: (i) there exists a Φ 0, ae L, such that a has property (*); (ii) there exists an algebra SI = {A; F} with (£(2t), the con- gruence lattice of 91, isomorphic to S, and there are α0, aί e A, α0 Φ alf such that (**) is satisfied for <α0, α:); (iii) /or α^?/ algebra §1 — <A; ί7) with (£(§I) isomorphic to S, ί/^βrβ are a0, a^A, a0 =£ a^ s^c/^ ί/^aί (**) is satisfied for <a0, ax> Let SΓ be the class of algebraic lattices having an a Φ 0 with property (*). Several simple observations can be made. The five element modular non-distributive lattice is not in 3Γ since none of the dual ideals generated by a nonzero element is prime. Every distributive algebraic lattice with a complete-join irreducible element is in 3ίΓ. If Si and S2 are algebraic lattices, then 21 + S2 e 192 WILLIAM A. LAMPE denotes ordinal sum). (The zero of S2 is a nonzero element in 8X + 82 having (*)). Every algebraic lattice 8 is both a complete sublattice of and a homomorphic image of a member of J%Γ since K^Se3ίΓm (G£Λ denotes the ^-element chain.) Also, observe that for a family (Si Ii 61) of algebraic lattices, 77(8;\ie I) e SΓ if and only if there exists at least one j el with Sy 6 3ίΓ. 2. The construction for representing (©,80,8,) as <@(2ί),@(2I),@(2t)>. First we need some notation. Let 21 = <A; F} be an algebra and X£ A. Set F(% X) = {φ\φ is an endomorphism of 21, {x} — xφ~ι for all xeX}, and set g(2I, X) = <F(2ί, X); o>. In other words, an endomorphism φ is in JF(2I, X) if (A — X)φ gi-I and xφ = x for x e X. Clearly, g(2t, X) is a nonempty semigroup with identity. S*(yf) is the subalgebra system of 21. Recall that Θ(2I) is the subalgebra lattice, that (£(21) is the congruence lattice, and that G?(2I) is the endomorphism semigroup. THEOREM 1. Suppose that 21 and 33 are algebras, that 21 is simple, that there is a £7g A, \U\ — 2, IT'S D for every DeS^{%), and that 2 there is an <α0, αL) e B with aQ Φ a1 for which property (**) is satis- fied. There exists an algebra 21' such that: (i ) @(2Γ) is isomorphic to @(2I); (ii) K(2ί') is isomorphic to (£(33); (iii) @(2ί') is isomorphic to g(St, [7). Proo/. Let 21 = <A; F> and 33 = <J5; (?> and U = {^0, %J and let 2 <α0, a^eB have (**) and let α0 ^ αlβ Assume that A and 5 are disjoint. For each x e B (J U define a nullary operation fx whose value is x.
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