Algebra Universalis最新文献

We introduce the category of Heyting frames, those coherent frames L in which the compact elements form a Heyting subalgebra of L, and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.

The collection of all cohereditary classes of modules over a ring R is a pseudocomplemented complete big lattice. The elements of its skeleton are the conatural classes of R-modules. In this paper we extend some results about cohereditary classes in R-Mod to the category (mathcal {L_{M}}) of linear modular lattices, which has as objects all complete modular lattices and as morphisms all linear morphisms. We introduce the big lattice of conatural classes in (mathcal {L_{M}}), and we obtain some results about it, paralleling the case of R-Mod and arriving at its being boolean. Finally, we prove some closure properties of conatural classes in (mathcal {L_{M}}).

In the paper, a basis of identities for the variety generated by the class of groupoids that are generalized subreducts of Tarski’s algebra of relations is found. It is also proved that the corresponding class of groupoids does not form a variety.

We investigate characterizations of the Galois connection ({{,textrm{Aut},}})-({{,textrm{sInv},}}) between sets of finitary relations on a base set A and their automorphisms. In particular, for (A=omega _1), we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under ({textrm{sInv Aut}}). Our structure (A, R) has an (omega )-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.

By studying the variety of Jónsson–Tarski algebras, we demonstrate two obstacles to the existence of large Jónsson algebras in certain varieties. First, if an algebra J in a language L has cardinality greater than (|L|^+) and a distributive subalgebra lattice, then it must have a proper subalgebra of size |J|. Second, if an algebra J in a language L satisfies ({{,textrm{cf},}}(|J|) > 2^{|L|^+}) and lies in a residually small variety, then it again must have a proper subalgebra of size |J|. We apply the first result to show that Jónsson algebras in the variety of Jónsson–Tarski algebras cannot have cardinality greater than (aleph _1). We also construct (2^{aleph _1}) many pairwise nonisomorphic Jónsson algebras in this variety, thus proving that for some varieties the maximum possible number of Jónsson algebras can be achieved.

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.