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Recently in [10, 11], the language of MV-algebras was extended by adding a unary operation, an internal operator, called also a state-operator. In [5], a stronger version of state MV-algebras, called state-morphism MV-algebras, was given. In this article, we present Stone Duality Theorems for (i) the category of Boolean algebras with a fixed state-operator and the category of compact Hausdorff topological spaces with a fixed idempotent continuous function, and for (ii) the category of weakly divisible σ-complete state-morphism MV-algebras and the category of Bauer simplices whose set of extreme points is basically disconnected and with a fixed idempotent continuous function. [ABSTRACT FROM PUBLISHER]

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding. [ABSTRACT FROM PUBLISHER]

We introduce, in this article, a partial order on quasi-morphic rings and investigate the behavior of regular elements in such rings. Further, the cleanness of regular elements in quasi-morphic rings having internal-cancelltion has been derived. [ABSTRACT FROM AUTHOR]

Tom Dieck introduced a commutative triangle whereby the exponential morphism from the Burnside functor to the unit functor is factorized through the real representation functor. Tornehave introduced a p-adic variant of the exponential morphism. His construction involves real representations that are not well defined up to isomorphism. To obtain a well defined commutative triangle, we introduce the orientation functor, a quotient of the real representation functor. [ABSTRACT FROM AUTHOR]

Banaschewski defines and gives sufficient conditions on a category A and a subclass M of its monomorphisms under which M-injectivity properly behaves with respect to the notions such as M-absolute retract, M-essentialness, and the existence of M-injective hulls. In this article, taking A to be the category of acts over a semigroup S and Md to be the class of sequentially dense monomorphisms, introduced and studied by Ebrahimi, Giuli, Mahmoudi, Moghaddasi, and Shahbaz, we study the behaviour of Md-injectivity (also called sequentially dense injectivity). We show, among other things, that S2 = S is a sufficient, but not necessary, condition for the behaviour of s-injectivity. Finally, we give an explicit description of the s-injective hulls of acts when they exist. [ABSTRACT FROM AUTHOR]

We study deformations of Leibniz algebra morphisms over a commutative local algebra base with 1. We construct the associated deformation cohomology that controls deformations using the cochain complex defining the Leibniz cohomology. [ABSTRACT FROM AUTHOR]

For a class L of right modules we investigate the relations of monomorphic L-envelopes and epimorphic L-covers over pullback-pushout diagrams, and obtain some isomorphism equations between the envelopes of covers and the covers of envelopes. [ABSTRACT FROM AUTHOR]

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding uA:A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (uA, uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are "u-extendable" is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ∀ B(frA(A, B) = U (A, B)), and, ∀ B ≠ 0(frA(B, A) = U(B, A)). [ABSTRACT FROM AUTHOR]

In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five. [ABSTRACT FROM AUTHOR]

If M and N are modules, the concept of semiregularity (and regularity) of hom ( M , N ) is defined and studied, and the connection with the relative direct injective- and direct projective-properties is established. The relationship of semiregularity to the Jacobson radical of hom ( M , N ), to the singular and cosingular ideals of hom ( M , N ), and to the notion of lying over or under a direct summand, is described, and the basic results in the module case are extended. Communicated by R. Wisbauer. [ABSTRACT FROM AUTHOR]

Let D be a Prüfer domain, and denote by ± b ℑ( D ) the multiplicative group of all invertible fractional ideals of D , ordered by A ≤ B if and only if A ⊇ B . Denote by G i the value group of the valuation associated with the valuation ring D M i , where { M i } i ∈ I is the collection of all maximal ideals of D . In this note we prove that the natural map from ± b ℑ( D ) into ± b ∏ i ∈ I G i is an isomorphism onto the cardinal sum ± b ∐ i ∈ I G i if and only if D is h -local. As a corollary, the group of divisibility of an h -local Bézout domain is isomorphic to ± b ∐ i ∈ I G i , the notation being as above. [ABSTRACT FROM AUTHOR]

After the proof of a not very known refinement of the Noether Normalization Theorem, we obtain two sharp degree bounds for the geometric degree of a dominating morphism of irreducible affine algebraic varieties and for the degree of the components of the inverse of an isomorphism of such varieties, generalizing by this way the well-known Gabber bound for automorphisms of affine spaces. [ABSTRACT FROM AUTHOR]

In this note,we answer a question of Hong et al. (2003) by proving that if a is a monomorphism of a reduced ring R , and R is a-skew Armendariz, then R is a-rigid. [ABSTRACT FROM AUTHOR]

Homological algebra based on fp-injective and fp-flat modules is studied. It can be realized as relative homological algebra corresponding to the proper class of monomorphisms Also, the corresponding homological functors Extf and Torf as well as various homological dimensions are investigated. [ABSTRACT FROM AUTHOR]

We describe the Lie isomorphisms on associative H*-algebras with zero annihilator. [ABSTRACT FROM AUTHOR]