Double semigroups have two associative operations ◦, • related by the interchange relation: (a • b) ◦ (c • d) ≡ (a ◦ c) • (b ◦ d). Kock [13] (2007) discovered a commutativity property in degree 16 for double semigroups: associativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with ...

In this paper we study the structure and the commutativity of a ring R, in which for each x,y ? R, there exist two integers depending on x,y such that [x,y]k equals x n or y n.

We extend the results of hep-th/0310137 to show that a general classical action for D = 2, N = 2 sigma models on a non(anti)commutative superspace is not standard and contains infinite number of terms, which depend on the determinant of the non(anti)commutativity parameter, C αβ. We show that using Kähler normal coordinates the action can be written in a manifestly covariant manner. We introduc...

We extend the results of hep-th/0310137 to show that a general classical action for D = 2, N = 2 sigma models on a non(anti)commutative superspace is not standard and contains infinite number of terms, which depend on the determinant of the non(anti)commutativity parameter, C αβ. We show that using Kähler normal coordinates the action can be written in a manifestly covariant manner. We introduc...

In this paper, we consider the finitely presented groups Gm and K(s, l) as follows; Gm = 〈a, b|a = b = 1, [a, b] = [a, b], [a, b] = [a, b]〉 K(s, l) = 〈a, b|ab = ba, ba = ab〉; and find the n-commutativity degree for each of them. Also we study the concept of n-abelianity on these groups, where m,n, s and l are positive integers, m,n ≥ 2 and g.c.d(s, l) = 1.