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.

Recently several generalizations to higher dimension of the classical Fourier transform (FT) using Clifford geometric algebra have been introduced, including the two-dimensional (2D) Clifford Fourier transform (CFT). Based on the 2D CFT, we establish the two-dimensional Clifford windowed Fourier transform (CWFT). Using the spectral representation of the CFT, we derive several important properti...

This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We giv...

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c0(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations ...

A hierarchy of segmentations(partitions) is a multiscale set representation of the image. This paper introduces a new set of scale space operators or transformations on the space of hierarchies of partitions. An ordering of hierarchies is proposed which is endowed by an ω-ordering based on a global energy over the classes of the hierarchy. A class of Matheron semigroups are shown to exists in t...

let s be a locally compact foundation semigroup with identity and                          be its semigroup algebra. let x be a weak*-closed left translation invariant subspace of    in this paper, we prove that  x  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. we also prove that a foundation semigroup with identity s is left amenab...

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.

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for the Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics [8]. The ZX calculus was known to be incomplete for quantum mechanics [7], on the other...

Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann algebras, complex numbers and quaternions. Lipschitz (1886) was the first to define groups constructed from ‘Clifford numbers’ and use them to represent rotations in a Euclidean space. É. Cartan discovered representations of the Lie algebras son(C) and son(R), n > 2, that do ...

In this paper, we attempt to investigate the connection between various types of ideals (for examples $(m, n)$-ideal, bi-ideal, interior ideal, quasi ideal, prime ideal and maximal ideal) of an ordered semigroup $(S,cdot ,leq)$ and the correspond hyperideals of its EL-hyperstructure $(S,*)$ (if exists). Moreover, we construct the class of EL-$Gamma$-semihypergroup associated to a partially orde...