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Any pentagonal quasigroup is proved to have the product , where is an Abelian group, is its regular automorphism satisfying and is the identity mapping. All Abelian groups of order inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is . We prove that the only translatable commutative pentagonal quasigroup is . The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group and its automorphism is proved to determine the value of and the range of values of

In this paper, for getting more results in groupoids, we consider a set and introduce the notion of a right (left) independent subset of a groupoid, and it is studied in detail. As a corollary of these properties, the following important result is proved: for any groupoid, there is a maximal right (left) independent subset. Moreover, the notion of strongly right (left) independent subset is considered. It is proved that there exists a groupoid having a strongly right independent 2-set. Finally, we discuss the notion of dynamic elements with independence.

Let X be a complete measure space of finite measure. The Lebesgue transform of an integrable function f on X encodes the collection of all the mean-values of f on all measurable subsets of X of positive measure. In the problem of the differentiation of integrals, one seeks to recapture f from its Lebesgue transform. In previous work we showed that, in all known results, f may be recaptured from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection A of all measurable subsets of X of positive measure. The first result of the present work is that the existence of such a limiting process is equivalent to the existence of a Von Neumann-Maharam lifting of X . In the second result of this work we provide an independent argument that shows that the recourse to filters is a necessary consequence of the requirement that the process of recapturing f from its mean-values is associated to a natural transformation , in the sense of category theory. This result essentially follows from the Yoneda lemma. As far as we know, this is the first instance of a significant interaction between category theory and the problem of the differentiation of integrals. In the Appendix we have proved, in a precise sense, that natural transformations fall within the general concept of homomorphism . As far as we know, this is a novel conclusion: Although it is often said that natural transformations are homomorphisms of functors, this statement appears to be presented as a mere analogy, not in a precise technical sense. In order to achieve this result, we had to bring to the foreground a notion that is implicit in the subject but has remained hidden in the background, i.e., that of partial magma .

The concept of a -translatable groupoid is explored in depth. Some properties of idempotent -translatable groupoids, left cancellative -translatable groupoids and left unitary -translatablegroupoids are proved. Necessary and sufficient conditions are found for a left cancellative -translatable groupoid to be a semigroup. Any such semigroup is proved to be left unitary and a union disjoint copies of cyclic groups of the same order. Methods of constructing -translatable semigroups that are not left cancellative are given.

We define and provide some basic analysis of various types of crossed products by semimultiplicative sets, and then prove a 𝐾𝐾-theoretical descent homomorphisms for semimultiplicative sets in accord with the descent homomorphism for discrete groups.

Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕ of a relation ϕ on X, defined by In particular, by using the relational inclusion ϕ ϕ with n, m ∈ , we show that the function α, defined by satisfies the Cauchy problem Moreover, the function f, defined by satisfies the translation problem Furthermore, the function F, defined by satisfies the Sincov problem Motivated by the above observations, we investigate a function F on the product set X to the power groupoid 𝒫(U) of an additively written groupoid U which is supertriangular in the sense that for all x, y, z ∈ X. For this, we introduce the convenient notations and Moreover, we gradually assume that U and F have some useful additional properties. For instance, U has a zero, U is a group, U is commutative, U is cancellative, or U has a suitable distance function; while F is nonpartial, F is symmetric, skew symmetric, or single-valued.

In this paper, we introduce a new algebra, called a BI-algebra, which is a generalization of a (dual) implication algebra and we discuss the basic properties of BI-algebras, and investigate ideals and congruence relations.

We prove that a pair of proximal Klee-Phelps convex groupoids (∘), (∘) in a finite-dimensional normed linear space are normed proximal, i.e., (∘) (∘) if and only if the groupoids are normed proximal. In addition, we prove that the groupoid neighbourhood (∘) ⊆ (∘) is convex in if and only if (∘) = (∘).

This paper aims to apply fuzzy sets and soft sets in combination to investigate algebraic properties of regular AG-groupoids. We initiate (∈γ; ∈γ qδ)-fuzzy soft left ideals (right ideals, bi-ideals and quasi-ideals) over AG-groupoids and explore some related properties. Moreover, we give a number of characterizations for regular AG-groupoids by virtue of various types of fuzzy soft ideals over them.