irreducible member
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2009 ◽  
Vol 88 (2) ◽  
pp. 231-238 ◽  
Author(s):  
KATE S. OWENS

AbstractA shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.



2004 ◽  
Vol 14 (03) ◽  
pp. 357-387 ◽  
Author(s):  
MARINA SEMENOVA ◽  
FRIEDRICH WEHRUNG

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by [Formula: see text] (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form [Formula: see text] where <Ti|i∈I> is a family of chains (resp., chains with at most n elements). We prove the following results: (1) Both classes [Formula: see text] and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equational bases of these varieties. (2) The variety [Formula: see text] is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of [Formula: see text]. (3) Every finite subdirectly irreducible member of [Formula: see text] is projective within [Formula: see text], and every subquasivariety of [Formula: see text] is a variety.



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