The variety DQD of semi-Heyting algebras with a weak negation, called dually quasi-De Morgan operation, and several of its subvarieties were investigated in the series [31], [32], [33], and [34]. In this paper we define and investigate a new subvariety JID of DQD, called “JI-distributive, dually quasi-De Morgan semi-Heyting algebras”, defined by the identity: x ∨ (y → z) ≈ (x ∨ y) → (x ∨ z), as...

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality...

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.

In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.

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...

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.