Categories and General Algebraic Structures with Applications
Volume:14 Issue: 1, Jan 2021

We study cocompleteness, co-wellpoweredness, and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness, and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.

In this article the notions of quasi mono (epi) as a generalization of mono (epi), (quasi weakly hereditary) general closure operator $mathbf{C}$ on a category $mathcal{X}$ with respect to a class $mathcal{M}$ of morphisms, and quasi factorization structures in a category $mathcal{X}$ are introduced. It is shown that under certain conditions, if $(mathcal{E}, mathcal{M})$ is a quasi factorization structure in $mathcal{X}$, then $mathcal{X}$ has a quasi right $mathcal{M}$-factorization structure and a quasi left $mathcal{E}$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $mathcal{M}$, every quasi factorization structure $(mathcal{E}, mathcal{M})$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $mathcal{M}$, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are provided.

We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of p-adic Hopf algebras. Through the duality, we introduce two sorts of p-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a p-adic Fourier transform compatible with the second p-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible p-adic unitary Banach representations of the open unit disc of the general linear group and its q-deformation in the case of dimension 2.

We define monoidal structures on several categories of linear topological modules over the valuation ring of a local field, and study module theory with respect to the monoidal structures. We extend the notion of the Iwasawa algebra to a locally profinite group as a monoid with respect to one of the monoidal structure, which does not necessarily form a topological algebra. This is one of the main reasons why we need monoidal structures. We extend Schneider--Teitelbaum duality to duality applicable to a locally profinite group through the module theory over the generalised Iwasawa algebra, and give a criterion of the irreducibility of a unitary Banach representation.

Let X be a zero-dimensional space and Cc(X) denote the functionally countable subalgebra of C(X). It is well known that β0X (the Banaschewski compactfication of X) is a quotient space of βX. In this article, we investigate a construction of β0X via βX by using Cc(X) which determines the quotient space of βX homeomorphic to β0X. Moreover, the construction of υ0X via υCc X (the subspace {p ∈ βX : ∀f ∈ Cc(X), f ∗ (p) < ∞} of βX) is also investigated.

If T is a semi-abelian algebraic theory, we prove that the category BornT of bornological T-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in BornT and, for a bornological T-algebra X, we investigate the relation between the representability of actions on X as a T-algebra and as a bornological Talgebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of BornT and prove in particular that both properties hold in the case of bornological groups.

In this paper, for a distributive lattice L, we study and compare some lattice theoretic features of L and topological properties of the Stone spaces Spec(L) and Max(L) with the corresponding graph theoretical aspects of the zero-divisor graph Γ(L). Among other things, we show that the Goldie dimension of L is equal to the cellularity of the topological space Spec(L) which is also equal to the clique number of the zero-divisor graph Γ(L). Moreover, the domination number of Γ(L) will be compared with the density and the weight of the topological space Spec(L). For a 0-distributive lattice L, we investigate the compressed subgraph ΓE(L) of the zero-divisor graph Γ(L) and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of L.

In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of Sheffer stroke and a Hilbert algebra. After it is shown that the axioms of Sheffer stroke Hilbert algebra are independent, it is given some properties of this algebraic structure. Then it is stated the relationship between Sheffer stroke Hilbert algebra and Hilbert algebra by defining a unary operation on Sheffer stroke Hilbert algebra. Also, it is presented deductive system and ideal of this algebraic structure. It is defined an ideal generated by a subset of a Sheffer stroke Hilbert algebra, and it is constructed a new ideal of this algebra by adding an element of this algebra to its ideal.