We show that a large number of equations are preserved by DedekindMacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that a large class of varieties of Heyting algebras and FL-algebras admits completions.

We give a complete description of subdirectly irreducible rings with involution satisfying x = x for some positive integer n. We also discuss ways to apply this result for constructing lattices of varieties of rings with involution obeying an identity of the given type. MSC 2000: 16W10, 08B26, 08B15

We study the semiring variety V generated by any finite number of finite fields F1, . . . , Fk and two-element distributive lattice B2, i.e., V = HSP{B2, F1, . . . , Fk}. It is proved that V is hereditarily finitely based, and that, up to isomorphism, B2 and all subfields of F1, . . . , Fk are the only subdirectly irreducible semirings in V.

As it is clearly suggested by the title, this note is a continuation of [1]. In the latter paper, the authors start from the famous theorem of N. Jacobson which asserts that every ring satisfying the identity x = x for some n ≥ 1 must be commutative (though Jacobson’s result is more general: the existence of a positive integer n(a) for each a ∈ R such that a = a suffices to conclude that the ri...

Recent studies of the algebraic properties of bilattices have provided insight into their internal structures, and have led to practical results, especially in reducing the computational complexity of bilattice-based multi-valued logic programs. In this paper the representation theorem for interlaced bilattices with negation found in 18] and extended to arbitrary interlaced bilattices without n...

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote’s negative solution to Problem 11 from [1]. R. McKenzie [2] has constructed a finitely generated variety having arbitrarily large finit...

Left distributive left quasigroups are binary algebras with unique left division satisfying the left distributive identity x(yz) ≈ (xy)(xz). In other words, binary algebras where all left translations are automorphisms. We provide a description and examples of non-idempotent subdirectly irreducible algebras in this class.

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...

The main goal of this paper is to introduce and investigate the related theory on monadic effect algebras. First, we design axiomatic system existential quantifiers algebras then use it give definition universal quantifier Then, relatively complete subalgebra prove that there exists a one-to-one correspondence between set all subalgebras. Moreover, characterize generated formula ideals Riesz co...