Derivations of Equality Algebras

In this paper, we introduced the concept of derivation on equality algebra E by using the notions of inner and outer derivations. Then we investigated some properties of (inner, outer) derivation and we introduced some suitable conditions that they help us to define a derivation on E. We introduced kernel and fixed point sets of derivation on E and prove that under which condition they are filters of E. Finally we prove that the equivalence relations on (E,⇝, 1) coincide with the equivalence relations on E with derivation d.(2010) MSC: 03G25, 06B10, 06B99.

Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations

<abstract><p>In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.</p></abstract>

Symmetries in Generalized Kaprekar’s Routine

In this paper, algebraic relations were established that determined the invariance of a transformed number after several transformations. The restrictions that determine the group structure of these relationships were analyzed, as was the case of the Klein group. Parametric Kr functions associated with the existence of cycles were presented, as well as the role of the number of their links in the grouping of numbers in higher-order equivalence classes. For this, we developed a methodology based on binary equivalence relations and the complete parameterization of the Kaprekar routine using Ki functions of parametric transformation.

Global aspects of measure preserving equivalence relations and graphs

This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view.

On Some Types of Multigranulation Covering Based on Binary Relations

Recently, the notions of right and left covering rough sets were constructed by right and left neighborhoods to propose four types of multigranulation covering rough set (MGCRS) models. These models were constructed using the granulations as equivalence relations. In this paper, we introduce four types of multigranulation covering rough set models under arbitrary relations using the q -minimal and q -maximal descriptors of objects in a given universe. We also study the properties of these new models. Thus, we explore the relationships between these models. Then, we put forward an algorithm to illustrate the method of reduction based on the presented model. Finally, we give an illustrative example to show its efficiency and importance.