CARL FAITH All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective . SISI rings include Noetherian rings, Morita rings, and almost maximal valuation rings ([Vil) . In [F3] we raised the question of whether a polynomial ring R[-1 over a SISI ring R is again SISI . In this paper we show this is not...

We prove that several problems concerning congruences on algebras are complete for nondeterministic log-space. These problems are: determining the congruence on a given algebra generated by a set of pairs, and determining whether a given algebra is simple or subdirectly irreducible. We also consider the problem of determining the smallest fully invariant congruence on a given algebra containing...

A semiring variety is d-semisimple if it is generated by the distributive lattice of order two and a finite number of finite fields. A d-semisimple variety V = HSP{B2, F1, . . . , Fk} plays the main role in this paper. It will be proved that it is finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1, . . . , Fk are the only subdirectly irr...

Abstract In this paper, we study the class of modules with fusion and implication based over distributive lattices , or FIDL - for short. We introduce concepts FIDL-subalgebra FIDL-congruence as well notions simple subdirectly irreducible FIDL-modules. give a bi-sorted Priestley-like duality FIDL-modules moreover, an application such duality, provide topological bi-spaced description FIDL-congr...

Let L1 be a finite simple sloop of cardinality n or the 8-element sloop. In this paper, we construct a subdirectly irreducible (monolithic) sloop L = 2⊗αL1 of cardinality 2n, for each n ≥ 8, with n ≡ 2 or 4 (mod 6), in which each proper homomorphic image is a Boolean sloop. Quackenbush [12] has proved that the variety V (L1) generated by a finite simple planar sloop L1 covers the smallest nontr...

Let A be a 4nitely generated variety of Heyting algebras and let SI(A) be the class of subdirectly irreducible algebras in A. We prove that A is dually equivalent to a category of functors from SI(A) into the category of Boolean spaces. The main tool is the theory of multisorted natural dualities. c © 2002 Elsevier Science B.V. All rights reserved. MSC: Primary: 06D20; 06D50; secondary: 08C05; ...

We give a representation of distributive lattices with the existential quantifier in terms of spectral spaces, which is an alternative to Cignoli’s representation in terms of Priestley spaces. Then we describe dual spectral spaces of subdirectly irreducible and simple Q-distributive lattices and prove that the variety QDist of Q-distributive lattices does not have the congruence extension prope...

We follow throughout a heterogeneous approach to present some characterizations (Theorem 3.5 and Theorem 3.35) of subdirectly irreducible bijective Boolean modules. The notions of essential element (as modular element) and [1]-closed element (as Boolean element) were implemented to this class of algebras. Our final aim of reaching the class of simple bijective Boolean modules, as a particular c...

We study clones on a four-element set related to the clone $${\mathsf {DMA}}$$ of all term functions subdirectly irreducible De Morgan algebra $${\mathbf {DM}}_{{{\mathbf {4}}}}$$ . find generating sets for preserving subalgebras , automorphisms truth order and information as well defined by conjunctions these conditions. identify covers in lattice four-valued describe above which contain discr...

The size of a concept lattice may increase exponentially with the size of the context. When the number of nodes is too large, it becomes very difficult to generate and study such a concept lattice. A way to avoid this problem is to break down the lattice into small parts. In the subdirect decomposition, the small parts are factor lattices which are meaningful in the Formal Concept Analysis (FCA...