Algebra Universalis最新文献

In this article we study the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.

The article proves topological representations for some classes of double Boolean algebras (dBas). In particular, representation theorems characterising fully contextual and pure dBas are obtained. Duality results for fully contextual and pure dBas are also established.

We show that any ordered group satisfying the identity ([x_1^{k_1}, ldots , x_n^{k_n}] = e) must be weakly abelian and that when (x_i not = x_1) for (2 le i le n), (ell )-groups satisfying the identity ([x_1^n, ldots , x_k^n] = e) also satisfy the identity ((x vee e)^{y^n} le (x vee e)^2). These results are used to study the structure of (ell )-groups satisfying identities of the form ([x_1^{k_1}, x_2^{k_2}, x_3^{k_3}] = e).

Let S be a multiplicatively idempotent congruence-simple semiring. We show that (|S|=2) if S has a multiplicatively absorbing element. We also prove that if S is finite then either (|S|=2) or (Scong {{,textrm{End},}}(L)) or (S^{op}cong {{,textrm{End},}}(L)) where L is the 2-element semilattice. It seems to be an open question, whether S can be infinite at all.

We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all four conditions of solvability, supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent.

A transformation of an ordered set M is an isotone mapping of M into M. By a representation of an ordered semigroup S by transformations of an ordered set M we mean a homomorphism of S into the set of transformations of M, i.e. (since the set of transformations of M is an ordered semigroup) an isotone mapping from S into the set of transformations of M preserving the operations. We prove that this type of representation leads to an “action” of S on M and so we introduce the notion of a left operand of M over S. Also we introduce the notions of a left operator pseudoorder on a left operand over S and a left operator homomorphism between left operands over S. We show that the concept of left operator pseudoorders on left operands over S plays an important role in the study of left operator homomorphisms of left operands over S. In the case of right operands over S dually definitions and results hold.

We show how modal quantales arise as convolution algebras (Q^X) of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and (Q^X). The catoids introduced generalise Schweizer and Sklar’s function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers.

A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere’s classical characterization of varieties of ordinary algebras. We also study the relationship of varieties to discrete Lawvere theories, and varieties as concrete categories over (mathbf{ Pos }).

This paper studies conditions in which a hyperarchimedean lattice-ordered group embeds into a hyperarchimedean lattice-ordered group with strong unit. While Conrad and Martinez showed that some hyperarchimedean lattice-ordered groups do not admit such embeddings, Section 3 presents a sufficient condition, in terms of the generalized Boolean algebra of principal (ell )-ideals, for the existence of such an embedding. Section 4 presents new examples of hyperarchimedean lattice-ordered groups not admitting such embeddings, while Section 5 shows that even when such an embedding exists, adjunction of a strong unit may yield non-isomorphic hyperarchimedean extensions. Section 6 shows that if one assumes the existence of weakly compact cardinals, then the sufficient condition from Section 3 is not necessary; and Section 7 studies the logical complexity of the condition “embeddable into a hyperarchimedean lattice-ordered group with strong unit.”