Algebra univers. 41 (1999) 229–2310002–5240/99/030229–03 $ 1.50+0.20/0© Birkha¨user Verlag, Basel, 1999MailboxA modification of Polin’s varietyK. A. KEARNES ANDM. VALERIOTE*Abstract. In this note we settle a question posed by Hobby and McKenzie in [2] on the nature of locallyfinite equational classes which satisfy some nontrivial congruence identity.We settle problem c 14 from [2] by exhibiting a locally finite equational class Vwhich omits types 1 and 5 and which satisfies some non-trivial congruence identity,but which contains a finite algebra having a type 4 minimal set with a nonemptytail. For background to this problem, the reader is encouraged to consult Chapter9 of [2] and the paper [1] by Day and Freese on Polin’s Variety.Loosely stated, Polin’s variety consists of algebras created by replacing thepoints of an (external) boolean algebra B by a family of (internal) boolean algebras{Ba: a B} so that whenever a]b in B, there is a homomorphism jbafrom Bato Bb.These homomorphisms are compatible in the sense that if a]b ]c then jca=jba$ jcb. The algebras come equipped with operations which allow one to recoverboth the internal and external boolean algebras.Our example is a simple modification of this idea wherein we replace theinternal boolean structures by distributive lattices having a distinguished largestelement 1. The proof that the resulting equational class satisfies some nontrivialcongruence identity is practically the same as that presented by Day and Freese forPolin’s variety in [1]. The tame congruence theoretic properties of the class referredto earlier can easily be established.Let B be a boolean algebra, S a function which assigns for each a B adistributive lattice S(a) with a largest element 1, and jba: S(a) S(b)a] b in Ba system of lattice homomorphisms such that jaa=idS(a)and if a] b] c thenjca=jcb$ jba.Presented by Professor Joel Berman.Received October 9, 1998; accepted in final form January 19, 1999.* Support of NSERC is gratefully acknowledged1991 Mathematics Subject Classification: Primary 08B05; Secondary 08A30.229K.A.KEARNES AND M.VALERIOTE230ALGEBRA UNIVERS.Define A(S, B) to be the algebra with universe bB{b}× S(b) and having thefollowing fundamental operations: (a, s) (b, t) =(a b, ja ba(s) ja bb(t)), (a, s)(b, t) =(a b, ja ba(s)ja bb(t)), (a, s)% =(a %, 1).Call such an algebra a boolean-by-distributive algebra, and let B/D be the closureunder isomorphism of the class of all such algebras. It is not difficult to see that thisclass is closed under subalgebras and direct products. Setting V to be theequational class generated by B/D we have:THEOREM 1 V is a locally finite equational class which satisfies the followingcongruence identity:x (yz)5 y(x(z(xy))). (1)Proof. The corresponding results for Polin’s variety are established in sections 2and 7 of [1]. The proofs presented there can be used, almost without change, toprove our theorem.THEOREM 2 V omits types 1 and 5 and contains a finite algebra A which hasa type 4 minimal set with nonempty tail.Proof. Since V satisfies a nontrivial congruence identity then by Theorem 9.18of [2], it omits types 1 and 5. As an exercise, the reader can check that V actuallyomits type 2 as well.Let A be the boolean-by-distributive algebra A(S, B) where B is the two elementboolean algebra, S(0) is a one element distributive lattice and S(1) is a two elementdistributive lattice. It can be easily checked that A has a unique nontrivialcongruence a, that the type of 0, a is 4 and that A is 0, a-minimal. This impliesthat the tail of A, considered as a 0, a-minimal set, is nonempty.Further generalizations of Polin’s construction have been investigated in themanuscript [3].REFERENCES[1] DAY, A. and FREESE, R., A characterization of identities implying congruence modularity, I., CanadianJournal of Mathematics32(1980), 1140–1167.A modification of Polin’s variety 231Vol. 41, 1999[2] HOBBY, D. and MCKENZIE, R., The Structure of Finite Algebras, volume 76 of ContemporaryMathematics. American Mathematical Society, 1988.[3] NEWRLY, N. and VALERIOTE, M., Some generalizations of Polin’s 6ariety, preprint, 1998.K. A. KearnesDepartment of MathematicsUni6ersity of Louis6illeLouis6ille, KY40292U.S.A.e-mail:[email protected]. ValerioteDepartment of Mathematics and StatisticsMcMaster Uni6ersityHamilton, Ontario L8S4K1Canadae-mail: