Picture Fuzzy Sets in Semigroups and Related Topics

Abstract

A picture fuzzy set is an extension of a classical fuzzy set. It contains a degree of membership, a degree of non-membership and a degree of neutral membership of each element x in a set X. In real-life situation, a picture fuzzy set is applicable in situations where human decisions demand more variety of responses: yes, abstain, no, and refusal. The concept of a picture fuzzy set was applied to many algebraic structures. In this research, we study a picture fuzzy set with a semigroup. First, we will define an interval-valued picture fuzzy subsemigroup and an interval-valued picture fuzzy left ideal [right ideal, ideal, bi-ideal, interior ideal, quasi-ideal, almost left ideal, almost right ideal, almost ideal, almost bi-ideal, almost interior ideal, almost quasi-ideal] of a semigroup.Next, we will investigate some properties of an interval-valued picture fuzzy subsemigroup and an interval-valued picture fuzzy left ideal [right ideal, ideal, bi-ideal, interior ideal, quasi-ideal, almost left ideal, almost right ideal, almost ideal, almost bi-ideal, almost interior ideal, almost quasi-ideal] of a semigroup. Moreover, some relationship between each ideal of a semigroup and its interval-valued picture fuzzification will be studied. Finally, we will study a picture N-fuzzy set and define a picture N-fuzzy left ideal [right ideal, ideal] of a semigroup. Furthermore, some properties of them in a semigroup are constructed.???????????????????????????????????????? ?????????????????????????????????? ???????????????????????????????????????????????????????? x ?????? X ????????????????????????????????????????????????????????????????????????? ???? ??????????????????????????????????????????????? ???????? ??????????? ?????????? ???????????????????? ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ????????????????????????????????? ?????????????????????????????? ????????????????????????????? ?????????????????????????? ??-?????????????????????????? ??????????????????????????????? ?????-?????????????????????????? ?????????????????????????????????????? ??????????????????????????????????? ?????????????????????????????????? ??????????????????????????????? ???????-?????????????????????????? ???????????????????????????????????? ?????????????-????????????????????????????????????? ?????????????????????????????????????????????????????????????????????????? ?????????????????????????????? ????????????????????????????? ?????????????????????????? ??-?????????????????????????? ??????????????????????????????? ?????-?????????????????????????? ?????????????????????????????????????? ??????????????????????????????????? ?????????????????????????????????? ??????????????????????????????? ???????-?????????????????????????? ???????????????????????????????????? ?????????????-????????????????????????????????????? ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ????????????????????????? ???????????????????????????????????? ????????????????????????????????????????????????????????