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Algebra univers 41 1999 229 231 0002 5240 99 030229 03 1 50 0 20 0 Birkha user Verlag Basel 1999 Mailbox A modification of Polin s variety K A KEARNES AND M VALERIOTE Abstract In this note we settle a question posed by Hobby and McKenzie in 2 on the nature of locally finite equational classes which satisfy some nontrivial congruence identity We settle problem c 14 from 2 by exhibiting a locally finite equational class V which 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 nonempty tail For background to this problem the reader is encouraged to consult Chapter 9 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 the points 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 j ab from Ba to Bb These homomorphisms are compatible in the sense that if a b c then j ac j ab j bc The algebras come equipped with operations which allow one to recover both the internal and external boolean algebras Our example is a simple modification of this idea wherein we replace the internal boolean structures by distributive lattices having a distinguished largest element 1 The proof that the resulting equational class satisfies some nontrivial congruence identity is practically the same as that presented by Day and Freese for Polin s variety in 1 The tame congruence theoretic properties of the class referred to earlier can easily be established Let B be a boolean algebra S a function which assigns for each a B a distributive lattice S a with a largest element 1 and j ab S a S b a b in B a system of lattice homomorphisms such that j aa idS a and if a b c then j ac j bc j ab Presented by Professor Joel Berman Received October 9 1998 accepted in final form January 19 1999 Support of NSERC is gratefully acknowledged 1991 Mathematics



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