Algebra Universalis最新文献

In this paper, new purely topological approaches are furnished in order to characterize self-majorizing elements in an Archimedean vector lattice A. More precisely, it is shown that an element (0<fin A) is a self-majorizing element if and only if every f-maximal order ideal of A is relatively uniformly closed. In addition, it is proved that self-majorizing elements are characterized via the hull–kernel topology on both the set of all proper prime order ideals ({mathcal {P}}) and on the set of all g-maximal order ideals ({mathcal {Q}}) of A, for all (gin A^{+}.) In fact, the set of all prime order ideals of A not containing f (respectively, the set of all g-maximal order ideals of A not containing f, for all (gin A^{+})) is a closed with respect to the hull–kernel topology on ({mathcal {P}}) (respectively, on ({mathcal {Q}})) if and only if f is a self-majorizing element in A.

We prove that slim patch lattices are exactly the absolute retracts with more than two elements for the category of slim semimodular lattices with length-preserving lattice embeddings as morphisms. Also, slim patch lattices are the same as the maximal objects L in this category such that (|L|>2.) Furthermore, slim patch lattices are characterized as the algebraically closed lattices L in this category such that (|L|>2.) Finally, we prove that if we consider ({0,1})-preserving lattice homomorphisms rather than length-preserving ones, then the absolute retracts for the class of slim semimodular lattices are the at most 4-element boolean lattices.

We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent.

We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then consider their minimal number of generators and show that it grows linearly with the depth of an associated rooted forest.

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.

Abstract

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.