Structure Of Semigroups Research Articles

The structure of ℒ-unipotent semigroups

The structure of ℒ-unipotent semigroups

Structure Mappings, Coextensions and Regular Four-Spiral Semigroups

The structure mapping approach to regular semigroups developed by K. S. S. Nambooripad and J. Meakin is used to describe the $\mathcal {K}$-coextensions of the fundamental four-sprial semigroup and hence to describe the structure of all regular semigroups whose idempotents form a four-spiral biordered set. Isomorphisms between regular four-spiral semigroups are studied. The notion of structural uniformity of a regular semigroup is defined and exploited.

On the classical content of many-body quantum mechanics

The aim of this paper is to reconcile the two modes of description of macrosystems, i.e., to remove certain inconsistencies between the classical phenomenological and the quantum-theoretical descriptions of a macrosystem. Starting from Ludwig's formulation of a general framework for classical theories and his ansatz for a compatibility condition between the quantum theoretical and the classical mode of description for a macrosystem, we try to make clear what the “classical content” of many-body quantum theory really is. It is shown that this classical content may be described by a certain “observable,” i.e., an operator-valued measure over the Borel sets of the classical trajectory space. There exists a reduced time evolution within this classical content which has the structure of a true semigroup in case of an irreversible classical time evolution.

Betweenness relations in probabilistic metric spaces

Four distinct versions of the betweenness concept for probabilistic metric are defined and studied. Conditions under which some or all of the properties of metric betweenness are satisfied are determined and the relationships among the different concepts are investigated. 1* Introduction. In his original paper on probabilistic metric [8] K. Menger, in addition to introducing the basic concepts and axioms, introduced a definition of betweenness, developed some of its properties and showed that this relation was generally weaker than ordinary metric betweenness. Shortly thereafter, A. [25] introduced a different definition of betweenness, based on a different triangle inequality, and showed that his relation did have all the properties of metric betweenness. Subsequently, J. F. C. Kingman [6] and F. Rhodes [16] studied betweenness in Wald spaces and H. Sherwood [23] considered a probabilistic version of the concept. Otherwise the subject has lain dormant—primarily because adequate tools for its analysis were not available. Our recent work on the structure of semigroups on the space of probability distribution [9, 11, 12, 17] and the development of characterist ic functions for certain classes of these semigroups [10, 13, 14] has changed this state of affairs. Thus we return to the study of betweenness in probabilistic metric spaces. We focus our attention on four different versions of this concept. The first of these is the straightforward generalization of Wald's betweenness from to arbitrary probabilistic metric spaces. We show that this relation satisfies some, but generally not all, of the usual properties of metric betweenness, determine sufficient conditions for the validity of those properties which are not always satisfied and show that in some instances these conditions are also necessary. The second betweenness relation applies to a restricted but nevertheless very large class of probabilistic metric spaces. It always satisfies the metric betweenness properties and, whenever it is comparable to the first relation, it is either identical or weaker. The Jhird relation, which applies to the same class of as the second, is obtained by an extension of Kingman's idea from to this class. It is a metric betweenness for certain naturally defined metrics and is always weaker than the second relation. The last relation is Menger's betweenness. We reformulate Menger's definition in terms of triangle and show that in

Structure mappings, coextensions and regular four-spiral semigroups

The structure mapping approach to regular semigroups developed by K. S. S. Nambooripad and J. Meakin is used to describe the K \mathcal {K} -coextensions of the fundamental four-sprial semigroup and hence to describe the structure of all regular semigroups whose idempotents form a four-spiral biordered set. Isomorphisms between regular four-spiral semigroups are studied. The notion of structural uniformity of a regular semigroup is defined and exploited.

On the structure of orthodox semigroups

On the structure of orthodox semigroups

Structure of regular semigroups. I

This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2

Correction to my paper on structure and ideal theory of commutative semigroups

Correction to my paper on structure and ideal theory of commutative semigroups

The fundamental four-spiral semigroup

Pastijn [17] showed that every semigroup may be embedded in a bisimple idempotent-generated semigroup. This resolves in the negative the conjecture of Eberhart, Williams, and Kinch [4] that a simple idempotent-generated semigroup is completely simple and suggests a study of the structure of bisimple idempotent-generated semigroups which are not completely simple. In this paper we introduce and analyse the structure of a semigroup which is a basic building block for bisimple non-completely simple idempotent-generated semigroups. This semigroup is fundamental (in the sense of Munn [13] and Nambooripad [15]) and has a biordered set of idempotents which can be described as a “spiral” biordered set: It is idempotent-generated, generated by four idempotents and can be described as a rectangular band of four semigroups, three of which are isomorphic to the bicyclic semigroup and one of which is a union of a bicyclic semigroup and an infinite cyclic semigroup.

The structure mappings on a regular semigroup

In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.

Conditionally Compact Semitopological One-Parameter Inverse Semigroups of Partial Isometries

The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if

The semigroup characterization of Osterwalder-Schrader path spaces and the construction of Euclidean fields

Axioms are given for relativistic quantum fields so the corresponding Schwinger functions are the expectation values of Euclidean fields. The main ingredient is the characterization of Osterwalder-Schrader path spaces by the associated semigroup structure.

A Generalization of Markov Processes

Osterwalder-Schrader (OS) positive symmetric stationary stochastic processes are discussed. A natural construction is given for the associated positive semigroup structure. Conversely, OS-positive symmetric stationary stochastic processes are constructed from positive semigroup structures. OS-positive processes are seen to be the natural generalization of Markov processes, positive semigroup structures being the natural generalization of positivity preserving semigroups. The inheritability of OS-positivity is discussed.

The semigroup of varieties of Brouwerian semilattices

It is shown that the semigroup of varieties of Brouwerian semilattices is free.

Conditionally compact semitopological one-parameter inverse semigroups of partial isometries

The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if B is the bicyclic semigroup and if B is contained in any semitopological semigroup, the relative topology on B is discrete. We show that if F is an inverse semigroup generated by an element and its inverse, and F is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if F is any one-parameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on F is jointly continuous if and only if the inversion is continuous on F, and we describe F ¯ \bar F in that case. We also show that if { J t } \{ {J_t}\} is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then { J t } ∪ { J t ∗ } \{ {J_t}\} \cup \{ J_t^\ast \} generates a one-parameter inverse semigroup T with J t − 1 = J t ∗ J_t^{ - 1} = J_t^\ast if and only if { J t } \{ {J_t}\} is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.

Structure and ideal theory of commutative semigroups

Structure and ideal theory of commutative semigroups

Semigroup structure in compactifications of ordered semigroups

Semigroup structure in compactifications of ordered semigroups

On the structure of orthodox semigroups

On the structure of orthodox semigroups

On the structure of inverse semigroups

On the structure of inverse semigroups

Algebraic implications of composability of physical systems

From classical and quantum mechanics we abstract the concept of a two-product algebra. One of its products is left unspecified; the other is a Lie product and a derivation with respect to the first. From composition of physical systems we abstract the concept of composition classes of such two-product algebras, each class being a semigroup with a unit. We show that the requirement of mutual consistency of the algebraic and the semigroup structures completely determines both the composition classes and the two-product algebras they consist of. The solutions are labelled by a single parameter which in the physical case is proportional to the square of the quantum of action.