We expand the notion of characteristic formula to infinite finitely presentable subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presentable subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of infinite algebras ...

A b s t r a c t. This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids (RSs). An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under finite joins. It is shown that a variety has this closure property if...

In [3] and also in [4], Freese and McKenzie prove that a finitely generated congruence modular variety is residually small if and only if it is residually less than n for some n < ~o. In fact, they showed that i fA generates a residually small congruence modular variety and [A] = m, then "//(A) is residually less than the bound l! + m where l = m . . . . 3. They obtained this result by associat...

A class TZ~ of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of 'V is decidable. Let y be a congruence modular variety. In this paper we prove that if "V is finitely decidable, then the following hold. (1) Each finitely generated subvariety of 2^ has a finite bound on the cardinality of its subdirectly irreducible members. (...

We give a characterization of the simple, and of the subdirectly irreducible boolean algebras with operators (including modal algebras), in terms of the dual descriptive frame. These characterizations involve a special binary quasi-reachability relation on the dual structure; we call a point u a quasi-root of the dual structure if every ultrafilter is quasireachable from u. We prove that a bool...

We study the structure of 0-primitive near-rings and are able to answer an open question in the theory of minimal ideals in near-rings to the negative, namely if the heart of a zero symmetric subdirectly irreducible near-ring is subdirectly irreducible again. Also, we will be able to classify when a simple near-ring with an identity and containing a minimal left ideal is a Jacobson radical near...

in this paper we study the notions of cogenerator and subdirectlyirreducible in the category of s-poset. first we give somenecessary and sufficient conditions for a cogenerator $s$-posets.then we see that under some conditions, regular injectivityimplies generator and cogenerator. recalling birkhoff'srepresentation theorem for algebra, we study subdirectlyirreducible s-posets and give this theo...