Algebra Universalis最新文献

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.”

Conditions are found on an operation on a finite set for its transform to be lacunary, that is, missing many expected terms. Often the condition is that the operation preserves a relation. General operations are split into odd and even lacunary parts, or more generally, into several lacunary parts given by a non-binary parity. This is applied to classical Fourier transforms as well as algebraic normal forms. With this theory, an explicit polynomial expansion is given for any operation in a preprimal algebra based on a finite elementary Abelian group. Convolution is defined, with a criterion given for it to commute with transforms.

We revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt’s recently proposed definition of bounded morphisms for polarities. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be regarded as fragments of sorted residuated (poly)modal logics, a research direction recently initiated by this author.

This paper is a contribution to understanding what properties should a topological algebra on a Stone space satisfy to be profinite. We reformulate and simplify proofs for some known properties using syntactic congruences. We also clarify the role of various alternative ways of describing syntactic congruences, namely by finite sets of terms and by compact sets of continuous self mappings of the algebra.

We answer two open problems about lattice-ordered groups that admit a connected lattice-ordered group topology. We show that, in the general case, admitting a connected lattice-ordered group topology does not effect the algebraic structure of the lattice-ordered group. For example, admitting a connected lattice-ordered group topology does not imply that the lattice-ordered group is Archimedean or even representable. On the other hand, if one assumes that the lattice-ordered group has a basis, then admitting a lattice-ordered group topology implies that the lattice-ordered group is a subdirect product of copies of the real numbers.

A ring is called a zip ring (Carl Faith coined this term) if every faithful ideal contains a finitely generated faithful ideal. By first proving that a reduced ring is a zip ring if and only if every dense element of the frame of its radical ideals is above a compact dense element, we study algebraic frames with the property stated in the title. We call them zipped. They generalize the coherent frames of radical ideals of zip rings, but (unlike coherent frames) they need not be compact. The class of zipped algebraic frames is closed under finite products, but not under infinite products. If the coproduct of two algebraic frames is zipped, then each cofactor is zipped. If the ring is not necessarily reduced, then its frame of radical ideals is zipped precisely when the ring satisfies what in the literature is called the weak zip property. For a Tychonoff space X, we show that C(X) is a zip ring if and only if X is a finite discrete space.