Algebra Universalis最新文献

Motivated by Haviar and Ploščica’s 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner’s Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs.

It is known that every effect algebra can be represented as the effect algebra of perspectivity classes of some E-test space. We describe when there exists join and meet of two perspectivity classes of events of some algebraic E-test space. Moreover we give the formula for join and meet of perspectivity classes mentioned above, using only tests. We obtain an example of finite, non-homogeneous effect algebra E such that sharp elements of E form a lattice, whereas E is not a lattice.

(mathfrak {KReg}) is the category of compact regular frames and frame homomorphisms. A class of (mathfrak {KReg}) frames (textbf{H}) is a hull class provided that: (i) (textbf{H}) is closed under isomorphic copies; (ii) for every (F in mathfrak {KReg}) there exist an (hF in textbf{H}) and a morphism (h_F) such that (F overset{h_F}{le } hF) is essential; (iii) if (F overset{phi }{le } H) is essential and (H in textbf{H}), then there exists (hphi : hF longrightarrow H) for which (phi = hphi cdot h_F). This work provides techniques for identifying and generating hull classes in (mathfrak {KReg}). Moreover, for a compact regular frame F, we introduce and investigate various properties of projectability and disconnectivity of F and prove that for each property, P, the class of (mathfrak {KReg})-objects that satisfy P is a hull class in (mathfrak {KReg}). In addition, we provide examples of (mathfrak {KReg}) hull classes that are not characterized by some form of projectability/disconnectivity and examples of classes of (mathfrak {KReg})-objects that are not hull classes.

In this paper, we solve two problems concerning the ideally conjunctive join-semilattices. First, we show that (L/R^1({{,textrm{Id},}}L)|_L) is ideally conjunctive for all join-semilattices L. Then we characterize those ideally conjunctive join-semilattices L such that ({{,textrm{coz},}}a) is compact for all (ain L.) Moreover, we give the definition of conjunctive posets and prove that the category of ideally conjunctive join-semilattices and join homomorphisms is reflective in the category of conjunctive posets and weakly ideal-continuous maps. As a corollary, we obtain the free ideally conjunctive join-semilattices over conjunctive posets.

The main goal of this paper is the study of the Boolean algebra of all characteristic elements in a unital (ell )-group and we investigate some topological properties of it in the case that the unital (ell )-group equipped with a link or positive filter topology. We also introduce the concept of Boolean region as a subset of a unital (ell )-group.

We continue our study of Sierpiński-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam’s theorem and its extension by Hajnal, it is proved that if (kappa ) is a regular uncountable cardinal that is not weakly compact in L, then there is a universal witness for non-weak-saturation of (kappa )-complete ideals. Specifically, there are (kappa )-many decompositions of (kappa ) such that, for every (kappa )-complete ideal J over (kappa ), and every (Bin J^+), one of the decompositions shatters B into (kappa )-many (J^+)-sets. A second focus here is the feature of narrowness of colourings, one already present in the theorem of Sierpiński. This feature ensures that a colouring suitable for an ideal is also suitable for all superideals possessing the requisite completeness degree. It is proved that unlike successors of regulars, every successor of a singular cardinal admits such a narrow colouring.

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

Abstract

C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. We characterize the bubble lattice both locally (via certain transformations of shuffle words) and globally (using a notion of inversion set). We then prove that the bubble lattice is extremal and constructable by interval doublings. Lastly, we prove that our bubble lattice is a generalization of the Hochschild lattice studied earlier by Chapoton, Combe and the second author.