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In this paper, we prove an isomorphism theorem for the case of refinement monoids with a group Γ acting on it. Based on this, we show a version of the well-known Jordan–Hölder theorem in this framework. The central result of this paper states that — as in the case of modules — a monoid T has a Γ -composition series if and only if it is both Γ -Noetherian and Γ -Artinian. As in module theory, these two concepts can be defined via ascending and descending chains, respectively. [ABSTRACT FROM AUTHOR]

This paper aims to study bounded algebras in another perspective-dual ideals of bounded L-algebras. As the dual concept of ideals in L-algebras, dual ideals are designed to characterize some significant properties of bounded L-algebras. We begin by providing a definition of dual ideals and discussing the relationships between ideals and dual ideals. Then, we prove that these dual ideals induce congruence relations and quotient L-algebras on bounded L-algebras. Naturally, in order to construct the first isomorphism theorem between bounded L-algebras, the relationship between dual ideals and morphisms between bounded L-algebras is investigated and that the kernels of any morphisms between bounded L-algebras are dual ideals is proven. Fortunately, although the first isomorphism theorem between arbitrary bounded L-algebras fails to be proven when using dual ideals, the theorem was proven when the range of morphism was good. Another main purpose of this study is to use dual ideals to characterize several kinds of bounded L-algebras. Therefore, first, the properties of dual ideals in some special bounded L-algebras are studied; then, some special bounded L-algebras are characterized by dual ideals. For example, a good L-algebra is a CL-algebra if and only if every dual ideal is C dual ideal is proven. [ABSTRACT FROM AUTHOR]

Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider function classes which besides the total functions only contain finite functions whose domain of definition is an initial segment of the natural numbers. Such functions appear naturally in computation. We show that a rich computability theory can be developed for these functions classes which embraces the central results of classical computability theory, in which all partial (computable) functions are considered. To do so, the concept of a Gödel number is generalised, resulting in a broader class of numberings. The central algorithmic idea in this approach is to search in enumerated lists. In this way, function computability is reduced to set listability. Besides the development of a computability theory for the functions classes, the new numberings – called quasi-Gödel numberings – are studied from a numbering-theoretic perspective: they are complete, and each of the function classes numbered in this way is a retract of the Gödel numbered set of all partial computable functions. Moreover, the Rogers semi-lattice of all computable numberings of the considered function classes is studied and results as in the case of the computable numberings of the partial computable functions are obtained. The function classes are shown to be effectively given algebraic domains in the sense of Scott–Ershov. The quasi-Gödel numberings are exactly the admissible numberings of the computable elements of the domain. Moreover, the domain can be computably mapped onto every other effectively given one so that every admissible numbering of the computable domain elements is generated by a quasi-Gödel numbering via this mapping. [ABSTRACT FROM AUTHOR]

Implicative semigroups with apartness have been introduced in 2016 by this author who then analyzed them in several papers. In this paper, a form of the third isomorphism theorem for this type of semigroups is shown, which has no counterpart in the classical semigroup theory. [ABSTRACT FROM AUTHOR]

The concept of a tolerance relation, shortly called tolerance, was studied on various algebras since the seventies of the twentieth century by B. Zelinka and the first author (see e.g. [6] and the monograph [1] and the references therein). Since tolerances need not be transitive, their blocks may overlap and hence in general the set of all blocks of a tolerance cannot be converted into a quotient algebra in the same way as in the case of congruences. However, G. Czédli ([7]) showed that lattices can be factorized by means of tolerances in a natural way, and J. Grygiel and S. Radeleczki ([8]) proved some variant of an Isomorphism Theorem for tolerances on lattices. The aim of the present paper is to extend the concept of a tolerance on a lattice to posets in such a way that results similar to those obtained for tolerances on lattices can be derived. [ABSTRACT FROM AUTHOR]

Let T(X) be the full transformation semigroup on the set X . For a fixed nonempty subset Y of X, let PGY (X) = {α ∈ T(X) : α|Y ∈ G(Y )} where G(Y ) is the permutation group on Y . It is known that P GY (X) is a regular subsemigroup of T(X). In this paper, we give a simpler description of Green’s relations and characterize the ideals of P GY (X). Moreover, we prove some isomorphism theorems for PGY (X). For finite sets, we investigate the cardinalities of PGY (X) and of its subsets of idempotents, and we also calculate their ranks. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce left almost seminearrings which generalize left almost semirings, near left almost rings and left almost rings. Some related properties of left almost seminearrings are investigated. Moreover, the ideal structure and its properties are studied. Finally, the isomorphism theorems are indicated. [ABSTRACT FROM AUTHOR]

This paper presents a Theory of Computation study based on recursive functions computability and innovates by performing parallels to relevant themes of Cryptography. Hence, it is presented the Hennie's notion of "abstract family of algorithms" (AFA, for short) according to the authors' understanding, and also more judicious theorems demonstrations, many times completely different from those ones available in literature. The main issue is the Isomorphism Theorem which supports the Church-Turing Thesis and provides a connection between Cryptology and Linguistics. [ABSTRACT FROM PUBLISHER]

Let T(X) be the full transformation semigroup on a nonempty set X. In this paper, the Cayley digraphs of T(X) with connection sets L and R, the Green's equivalence classes of T(X) according to the Green's relations L and R, are investigated. Furthermore, their connectedness properties are characterized. In addition, the isomorphism theorems for Cayley digraphs of T(X) are also presented.

Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R-hypermodules with inclusion single-valued R-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner R-hypermodules with strong single-valued R-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued R-homomorphism f : A → B and a ∈ A , we have K e r (f) + a = a + K e r (f) = { x ∈ A ∣ f (x) = f (a) } . [ABSTRACT FROM AUTHOR]

In this paper, the concepts of multiplication (or addition) extension of hyperlattices are introduced, and the construction of transposition hyperlattices are presented. The cosets of hyperlattices and the quotient structures of hyperlattices are constructed. Some properties of quotient hyperlattices are studied. Moreover the First, Second and Third Isomorphism Theorems are established. [ABSTRACT FROM AUTHOR]

Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R-hypermodules with inclusion single-valued R-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner R-hypermodules with strong single-valued R-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued R-homomorphism f : A → B and a ∈ A , we have K e r ( f ) + a = a + K e r ( f ) = { x ∈ A ∣ f ( x ) = f ( a ) } .

The notion of a multi-term π-institution is introduced and a criterion for the equivalence of two multi-term π-institutions in terms of their categories of theories is proved. Moreover, a counterexample that shows that this criterion is false for arbitrary π-institutions is given. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]