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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]

The isomorphism theorems for crossed squares of commutative algebras, which arise when the crossed modules of algebras are given an extra dimension, are the main subject of this paper. The definition of crossed squares of commutative algebras is given in this context, encompassing ideas like the crossed square ideal, image, and quotient crossed squares, as well as the kernel for crossed square morphisms. The study discusses the way how isomorphism theorems are applied to these structures and offers detailed proofs for this framework. Moreover, some necessary concepts such as quotient crossed squares, which were not previously specified in these structures, are also presented, and some basic properties are examined. The study provides opportunities for possible generalization to a number of different structures, including crossed n-cubes. [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 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.