Algebra Universalis最新文献

We prove that the modal logic of abelian groups with the accessibility relation of being isomorphic to a subgroup is (mathsf {S4.2}).

We establish a criterion for a structure M on an infinite domain to have the Galois closure ({{,textrm{InvAut},}}(M)) (the set all relations on the domain of M that are invariant to all automorphisms of M) defined via infinite Boolean combinations of infinite (constructed by infinite conjunction) existential relations from M. Based on this approach, we present criteria for quantifier elimination in M via finite partial automorphisms of all existential relations from M, as well as criteria for (weak) homogeneity of M. Then we describe properties of M with a countable signature, for which the set of all relations, expressed by quantifier-fee formulas over M, is weakly inductive, that is, this set is closed under any infinitary intersection of the same arity relations. It is shown that the last condition is equivalent: for every (n ge 1) there are only finitely many isomorphism types for substructures of M generated by n elements. In case of algebras with a countable signature such type can be defined by the set of all solutions of a finite system of equations and inequalities produced by n-ary terms over those algebras. Next, we prove that for a finite M with a finite signature the problem of the description of any relation from ({{,textrm{InvAut},}}(M)) via the first order formula over M, which expresses it, is algorithmically solvable.

In this paper, the relationships between two important subclasses of algebraic dcpos and topological spaces which may not be (textrm{T}_0) are discussed. The concepts of CFF-spaces and strong CFF-spaces are introduced by considering the properties of their topological bases. With these concepts, Lawson compact algebraic L-domains and Scott domains are successfully represented in purely topological terms. Moreover, equivalences of the categories corresponding to these two subclasses of algebraic dcpos are also provided. This opens a way of finding non-(textrm{T}_0) topological characterizations for domains.

In this paper, we characterize when, for any infinite cardinal (alpha ), the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind (alpha )-complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself.

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term (theta )). These generalise weakly Schreier extensions of monoids, as well as general extensions of semi-abelian varieties (using the (theta ) appearing in their syntactic characterisation). Restricting again to the case of monoids, a different choice of (theta ) leads to a new class of monoid extensions, more general than the weakly Schreier split extensions.

Each strongly minimal Steiner k-system (M, R) (where is R is a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by a quasigroup if k is a prime-power. We show this coordinatization is never definable in (M, R) and the strongly minimal Steiner k-systems constructed in (Baldwin–Paolini 2020) never interpret a quasigroup. Nevertheless, by refining the construction, if k is a prime power, in each (2, k)-variety of quasigroups (Definition 3.10) there is a strongly minimal quasigroup that interprets a Steiner k-system.

We consider ordered universal algebras and give a construction of a join-completion for them using so-called (mathscr {D})-ideals. We show that this construction has a universal property that induces a reflector from a certain category of ordered algebras to the category of sup-algebras. Our results generalize several earlier known results about different ordered structures.

Stone space partitions ({X_{p}mid pin P}) satisfying conditions like (overline{X_{p}}=bigcup _{qleqslant p}X_{q}) for all (pin P), where P is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of (omega )-categorical structures. A key concept for studying such partitions is that of a p-trim open set which meets precisely those (X_{q}) for which (qgeqslant p); for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition ({X_{r}mid rin [0,1]}) of the Cantor set such that (overline{X_{r}}=bigcup _{sleqslant r}X_{s}text { for all }rin [0,1]).

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system (langle X rangle _P). The closure system (langle Y rangle _P) generated by the patch closure Y of X is the patch closure of (langle X rangle _P). If X is contained in the set of nontrivial prime elements of P then (langle X rangle _P) is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.