In this paper, a commutative semigroup will be written as a disjoint union of its cancellative subsemigroups. Based on this fact we will define the left regular representation of a commutative separative semigroup and show that this representation is faithful. Finally concrete examples of commutative separative semigroups, their decompositions and their left regular representations are given.

A reductive Q-simple algebraic group G is of hermitian type, if the symmetric space D defined by G(R) is a hermitian symmetric space. A discrete subgroup Γ ⊂ G(Q) is arithmetic, if for some faithful rational representation ρ : G −→ GL(V ), and for some lattice VZ ⊂ VQ, Γ is commensurable with ρ−1(GL(VZ)). A non-compact hermitian symmetric space is holomorphically equivalent to a bounded symmetr...

‎‎let r be a non-commutative ring with unity‎. ‎the commuting graph‎ ‎of $r$ denoted by $gamma(r)$‎, ‎is a graph with a vertex set‎ ‎$rsetminus z(r)$ and two vertices $a$ and $b$ are adjacent if and only if‎ ‎$ab=ba$‎. ‎in this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. it is shown that‎, ‎$gamma(r)$ is the disjoint ...

A group–category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1–dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group–categories, in hopes of finding clues to a better u...

In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical light-cone str...

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen–Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred...

let $r$ be a commutative ring. in this paper, by using algebraic properties of $r$, we study the hase digraph of prime ideals of $r$.