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This study introduces an ample Hausdorff groupoid Aˆ ⋊ R extracted from an ample Hausdorff groupoid G and a unital commutative ring R; a Hausdorff groupoid D which is the discrete twist over Aˆ ⋊ R. In the groupoid C*-algebra perspective, when R = C there is an isomorphism between the non-twisted groupoid C*-algebra (C∗(G)) and the twisted groupoid C*-algebra (C∗(Aˆ ⋊ R;D)). However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra (AR(G)) and the twisted Steinberg algebra (AR(D;Aˆ⋊R)) are non-isomorphic. [ABSTRACT FROM AUTHOR]

In the present paper, we apply the concept of reverse derivation in rings on the concept of d - algebra to obtain the concept called a left-right (resp. right-left) reverse derivations of d-algebra X (briefly, (l, r) resp. (r, l)-reverse derivation of d-algebra), we will also, define some concepts such as regular map, composition two maps and study the related properties. Moreover, the notions of partial ordered edge d - algebra as well as d - subalgebra and their relation to our current study are obtained. In addition, some illustrative examples and counterexamples are discussed. [ABSTRACT FROM AUTHOR]

A pseudo hyper GR-algebra is an algebraic structure involving two distinct hyperoperations. Properties of this hyper algebra have been studied and given illustrations. This paper focuses on the quotient structure of pseudo hyper GR-algebras. From an equivalence relation on a pseudo hyper GR-algebra H, we can define a congruence relation on H that is used in the construction of the quotient structure H/I, where I is the congruence class of 0 under the congruence relation. Moreover, some isomorphism theorems of pseudo hyper GR-algebras are included in this paper. [ABSTRACT FROM AUTHOR]

In this paper we study the class of algebras satisfying a polynomial identity of degree six that are principal train algebras of rank 3 or 4, for which we give the explicit form of the train equation. If the rank of A is n ≥ 5 in general, we provide the form of the train equation in some cases. [ABSTRACT FROM AUTHOR]

This paper is devoted to the classification of train algebras of degree 2 and exponent 4. This classification is made in dimension at most four and according to the type of the algebra. We first show that in four dimension, the type of algebra can only be of (2, 0, 1, 1), (2, 1, 0, 1) or (2, 1, 1, 0). [ABSTRACT FROM AUTHOR]

This paper introduces the fuzzification of the dual B-algebra and presented some of its related properties. A characterization of a fuzzy (normal) dual B-algebra is also investigated. [ABSTRACT FROM AUTHOR]

The concept of the direct product of finite family of B-algebras is introduced by Lingcong and Endam[J. A. V. Lingcong and J. C. Endam, Direct product of B-algebras, Int. J. Algebra, 10(1):33-40, 2016.]. In this paper, we introduce the concept of the direct product of infinite family of B-algebras, we call the external direct product, which is a generalization of the direct product in the sense of Lingcong and Endam. Also, we introduce the concept of the weak direct product of B-algebras. Finally, we provide several fundamental theorems of (anti-)B-homomorphisms in view of the external direct product B-algebras. [ABSTRACT FROM AUTHOR]