Some remarks on the four-spiral semigroup.
In this paper, we introduce the concept of bipolar-valued fuzzification of ordered 𝓐𝓖-groupoids and discuss some structural properties of bipolar-valued fuzzy two-sided ideals of an intra-regular ordered 𝓐𝓖-groupoid.
In an earlier paper, the second author generalized Eilenberg's variety theory by establishing a basic correspondence between certain classes of monoid morphisms and families of regular languages. We extend this theory in several directions. First, we prove a version of Reiterman's theorem concerning the definition of varieties by identities, and illustrate this result by describing the identities associated with languages of the form (a1a2...ak)+, where a1,...,ak are distinct letters. Next,...
We study the group structure in terms of the number of Sylow -subgroups, which is denoted by . The first part is on the group structure of finite group such that , where is a normal subgroup of . The second part is on the average Sylow number and we prove that if is a finite nonsolvable group with and , then , where is the Fitting subgroup of . In the third part, we study the nonsolvable group with small sum of Sylow numbers.