Algebra Universalis最新文献

We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form (f(x_1,dots ,x_n)approx g(y_1,dots ,y_m)), also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the ({text {CSP}}) of a finite structure (mathbb {A}) only depends on the set of minor identities satisfied by the polymorphism clone of (mathbb {A}). In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write (mathcal {C} {preceq _{textrm{m}}} mathcal {D}) if there exists a minor homomorphism from (mathcal {C}) to (mathcal {D}). We show that the aforementioned poset has only three submaximal elements.

The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine k-ary parts of the clones and their mutual inclusions. In this paper we study k-ary parts of maximal clones, for (kgeqslant 2,) building on the already known results for their unary parts. It turns out that the poset of k-ary parts of maximal clones defined by central relations contains long chains.

Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) (varrho ) have the property that an n-ary operation f preserves (varrho ,) i.e., f is a polymorphism of (varrho ,) if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves (varrho ,) i.e., it is an endomorphism of (varrho .) We introduce a wider class of relations—called generalized quasiorders—of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection ({textrm{End}})–({{,textrm{gQuord},}},) i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.

Inspired by locale theory, pointfree convex geometry was first proposed and studied by Yoshihiro Maruyama. In this paper, we shall continue to his work and investigate the related topics on pointfree convex spaces. Concretely, the following results are obtained: (1) A Hofmann–Lawson-like duality for pointfree convex spaces is established. (2) The (mathcal {M})-injective objects in the category of (S_0)-convex spaces are proved precisely to be sober convex spaces, where (mathcal {M}) is the class of strict maps of convex spaces; (3) A convex space X is sober iff there never exists a nontrivial identical embedding (i:Xhookrightarrow Y) such that its dualization is an isomorphism, and a convex space X is (S_D) iff there never exists a nontrivial identical embedding (k:Yhookrightarrow X) such that its dualization is an isomorphism. (4) A dual adjunction between the category (textbf{CLat}_D) of continuous lattices with continuous D-homomorphisms and the category (textbf{CS}_D) of (S_D)-convex spaces with CP-maps is constructed, which can further induce a dual equivalence between (textbf{CS}_D) and a subcategory of (textbf{CLat}_D); (5) The relationship between the quotients of a continuous lattice L and the convex subspaces of ({textbf {cpt}}(L)) is investigated and the collection ({textbf {Alg}}({textbf {Q}}(L))) of all algebraic quotients of L is proved to be an algebraic join-sub-complete lattice of ({textbf {Q}}(L)) of all quotients of L, where ({textbf {cpt}}(L)) denote the set of non-bottom compact elements of L. Furthermore, it is shown that ({textbf {Alg}}({textbf {Q}}(L))) is isomorphic to the collection ({textbf {Sob}}(mathcal {P}({textbf {cpt}}(L)))) of all sober convex subspaces of ({textbf {cpt}}(L)); (6) Several necessary and sufficient conditions for all convex subspaces of ({textbf {cpt}}(L)) to be sober are presented.

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.