Categories and General Algebraic Structures with Applications

Let RL be the ring of real-valued continuous functions on a frame L as the pointfree version of C(X), the ring of all real-valued continuous functions on a topological space X. Since Cc(X) is the largest subring of C(X) whose elements have countable image, this motivates us to present the pointfree version of Cc(X). The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in RL. In particular , we will introduce the pointfree version of the ring Cc(X). We define a relation from RL into the power set of R, namely overlap. Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as Im(f) ⊆ S for every continuous function f : X → R and S ⊆ R.

For a frame L, consider the f-ring F_PL=Frm(P(R),L). In this paper, first we show that each minimal ideal of F_PL
is a principal ideal generated by f_a, where a is an atom of L. Then we show that if L is an F_P-completely regular frame, then the socle of F_PL consists of those f for which coz(f) is a join of finitely many atoms. Also it is shown that not only F_PL has Property (A) but also if L
has a finite number of atoms then the residue class ring
F_PL/Soc(F_PL) has Property (A).

This paper is devoted to the study of products of classes of right S-posets possessing one of the flatness properties and preservation of such properties under products. Specifically, we characterize a pomonoid S over which its nonempty products as right S-posets satisfy some known flatness properties. Generalizing this results, we investigate products of right S-posets satisfying Condition (P W P). Finally, we investigate pomonoids over which products of right S-posets transfer an arbitrary flatness property, projectivity, freeness, and regularity to their components.

Laan introduced the principal weak form of Condition (P) as Condition (P W P) and gave some characterization of monoids by this condition of their acts. In this paper first we introduce Condition (G-P W P), a generalization of Condition (P W P) of acts over monoids and then will give a characterization of monoids when all right acts satisfy this condition. We also give a characterization of monoids, by comparing this property of their acts with some others. Finally, we give a characterization of monoids coming from some special classes, by this property of their diagonal acts and extend some results on Condition (P W P) to this condition of acts.

Let G = (V, E) be a graph. A subset S of V is a dominating set of G if every vertex in V \ S is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for each v ∈ V \ S there exists u ∈ S such that v is adjacent to u and S1 = (S \ {u}) ∪ {v} is a dominating set. If further the vertex u ∈ S is unique, then S is called a perfect secure dominating set. The minimum cardinality of a perfect secure dominating set of G is called the perfect secure domination number of G and is denoted by γ ps (G). In this paper we initiate a study of this parameter and present several basic results.

In this paper, for each lattice-valued map A → L with some properties, a ring representation A → RL is constructed. This representation is denoted by τc which is an f-ring homomorphism and a Q-linear map, where its index c, mentions to a lattice-valued map. We use the notation δ a pq = (a − p) + ∧ (q − a) + , where p, q ∈ Q and a ∈ A, that is nominated as interval projection. To get a well-defined f-ring homomorphism τ c , we need such concepts as bounded, continuous, and Q-compatible for c, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map c ϕ : A → L for each f-ring homomorphism ϕ : A → RL. It is proved that cτ c = c r and τc ϕ = ϕ, which they make a kind of correspondence relation between ring representations A → RL and the lattice-valued maps A → L, where the mapping c r : A → L is called a realization of c. It is shown that τ c r = τ c and c rr = c r. Finally, we describe how τc can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.

In this article we investigate filters of cozero sets for real-valued continuous functions, called coz-filters. Much is known for z-ultrafilters and their correspondence with maximal ideals of C(X). Similarly, a correspondence will be established between coz-ultrafilters and minimal prime ideals of C(X). We will further notice various properties of coz-ultrafilters in relation to P-spaces and F-spaces. In the last two sections, the collection of coz-ultrafilters will be topologized, and then compared to the hull-kernel and the inverse topologies placed on the collection of minimal prime ideals of C(X) and general lattice-ordered groups.

Equality algebras were introduced by S. Jenei as a possible algebraic semantic for fuzzy type theory. In this paper, we introduce some types of filters such as (positive) implicative, fantastic, Boolean, and prime filters in equality algebras and we prove some results which determine the relation between these filters. We prove that the quotient equality algebra induced by an implicative filter is a Boolean algebra, by a fantastic filter is a commutative equality algebra, and by a prime filter is a chain, under suitable conditions. Finally, we show that positive implicative, implicative, and Boolean filters are equivalent on bounded commutative equality algebras.

In this paper, the main results are: a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I; a study of Hopfian MV-algebras; and a category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism).

Cozero maps are generalized forms of cozero elements. Two particular cases of cozero maps, slim and regular cozero maps, are significant. In this paper we present methods to construct slim and regular cozero maps from a given cozero map. The construction of the slim and the regular cozero map from a cozero map are called slimming and regularization of the cozero map, respectively. Also, we prove that the slimming and regularization create reflector functors, and so we may say that they are the best method of constructing slim and regular cozero maps, in the sense of category theory. Finally, we give slim regularization for a cozero map c : M → L in the general case where A is not a Q-algebra. We use the ring and module of fractions, in this construction process.

Functionals were discovered and used by Volterra over a century ago in his study of the motions of viscous elastic materials and electromagnetic fields. The need to precisely account for the qualitative effects of the cohesion and shape of the domains of these functionals was the major impetus to the development of the branch of mathematics known as topology, and today large numbers of mathematicians still devote their work to a detailed technical analysis of functionals. Yet the concept needs to be understood by all people who want to fully participate in 21st century society. Through some explicit use of mathematical categories and their transformations, functionals can be treated in a way which is non-technical and yet permits considerable reliable development of thought. We show how a deformable body such as a storm cloud can be viewed as a kind of space in its own right, as can an interval of time such as an afternoon; the infinite-dimensional spaces of configurations of the body and of its states of motion are constructed, and the role of the infinitesimal law of its motion revealed. We take nilpotent infinitesimals as given, and follow Euler in defining real numbers as ratios of infinitesimals.