scholarly journals Naturally ordered regular semigroups with maximum inverses

1989 ◽  
Vol 32 (1) ◽  
pp. 33-39 ◽  
Author(s):  
Tatsuhiko Saito
Keyword(s):  
Regular Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Inverse Subsemigroup ◽  
Necessary And Sufficient

Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.

scholarly journals Idempotent-separating extensions of regular semigroups

2005 ◽  
Vol 2005 (18) ◽  
pp. 2945-2975
Author(s):  
A. Tamilarasi
Keyword(s):  
Automorphism Group ◽  
Regular Semigroup ◽  
Extension Theory ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compatibility Conditions ◽  
Regular Semigroups ◽  
Biordered Set ◽  
Crossed Pair ◽  
Necessary And Sufficient

For a regular biordered setE, the notion ofE-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered setE, anE-diagram in a categoryCis a collection of objects, indexed by the elements ofEand morphisms ofCsatisfying certain compatibility conditions. With such anE-diagramAwe associate a regular semigroupRegE(A)havingEas its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application ofRegE(A)to the idempotent-separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a groupE-diagramA. In this paper, the necessary and sufficient condition for the existence of an extension ofSbyAis provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.

Regular semigroups with a multiplicative inverse transversal

1982 ◽  
Vol 92 (3-4) ◽  
pp. 253-270 ◽  
Author(s):  
T. S. Blyth ◽  
R. McFadden
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Multiplicative Inverse ◽  
Multiplicative Property ◽  
Regular Semigroups ◽  
Inverse Subsemigroup

SynopsisBy an inverse transversal of a regular semigroup S we mean an inverse subsemigroup that contains a single inverse of every element of S. A certain multiplicative property (which in the case of a band is equivalent to normality) is imposed on an inverse transversal and a complete description of the structure of S is obtained.

Perfect Dubreil-Jacotin semigroups

1977 ◽  
Vol 78 (1-2) ◽  
pp. 101-104 ◽  
Author(s):  
T. S. Blyth
Keyword(s):  
Regular Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

SynopsisWe determine a set of conditions that are necessary and sufficient for an ordered regular semigroup to be a perfect Dubreil-Jacotin semigroup; and a necessary and sufficient condition for a perfect Dubreil-Jacotin semigroup to be orthodox.

scholarly journals On right self-injective regular semigroups, II

1983 ◽  
Vol 34 (2) ◽  
pp. 182-198 ◽  
Author(s):  
Kunitaka Shoji
Keyword(s):  
Extension Property ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Necessary And Sufficient Condition ◽  
Regular Semigroups ◽  
Left Congruence ◽  
Spined Product ◽  
The Right ◽  
Necessary And Sufficient

AbstractIt is shown that a semigroup is right self-injective and a band of groups if and only if it is isomorphic to the spined product of a self-injective semilattice of groups and a right self-injective band. A necessary and sufficient condition for a band to be right self-injective is given. It is shown that a left [right] self-injective semigroup has the [anti-] representation extension property and the right [left] congruence extension property.

scholarly journals Construction of regular semigroups with inverse transversals

1989 ◽  
Vol 32 (1) ◽  
pp. 41-51 ◽  
Author(s):  
Tatsuhiko Saito
Keyword(s):  
Regular Semigroup ◽  
Unique Element ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Inverse Subsemigroup

Let S be a regular semigroup. An inverse subsemigroup S° of S is an inverse transversal if |V(x)∩S°| = 1 for each x∈S, where V(x) denotes the set of inverses of x. In this case, the unique element of V(x)∩S° is denoted by x°, and x°° denotes (x°)–1. Throughout this paper S denotes a regular semigroup with an inverse transversal S°, and E(S°) = E° denotes the semilattice of idempotents of S°. The sets {e∈S:ee° = e} and {f∈S:f°f=f} are denoted by Is and Λs, respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S. When both I and Λ are sub-bands of S, S° is called an S-inverse transversal. An inverse transversal S° is multiplicative if x°xyy°∈E°, and S° is weakly multiplicative if (x°xyy°)°∈E° for every x, y∈S. A band B is left [resp. right] regular if e f e = e f [resp. e f e = f e], and B is left [resp. right] normal if e f g = e g f [resp. e f g = f e g] for every e, f, g∈B. A subset Q of S is a quasi-ideal of S if QSQ ⊆ S.

scholarly journals Regularity in terms of Hyperideals

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Kostaq Hila ◽  
Bijan Davvaz ◽  
Krisanthi Naka ◽  
Jani Dine
Keyword(s):  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Systems ◽  
Regular Semigroups ◽  
Nilpotent Elements ◽  
Necessary And Sufficient ◽  
Regular Rings

This paper deals with the algebraic hypersystems. The notion of regularity of different type of algebraic systems has been introduced and characterized by different authors such as Iseki, Kovacs, and Lajos. We generalize this notion to algebraic hypersystems giving a unified generalization of the characterizations of Kovacs, Iseki, and Lajos. We generalize also the concept of ideal introducing the notion of j-hyperideal and hyperideal of an algebraic hypersystem. It turns out that the description of regularity in terms of hyperideals is intrinsic to associative hyperoperations in general. The main theorem generalizes to algebraic hypersystems some results on regular semigroups and regular rings and expresses a necessary and sufficient condition by means of principal hyperideals. Furthermore, two more theorems are obtained: one is concerned with a necessary and sufficient condition for an associative, commutative algebraic hypersystem to be regular and the other is concerned with nilpotent elements in the algebraic hypersystem.

scholarly journals Free generators in free inverse semigroups: Corrigenda

1973 ◽  
Vol 9 (3) ◽  
pp. 479-480 ◽  
Author(s):  
N.R. Reilly
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Subsemigroup ◽  
Free Generators ◽  
Necessary And Sufficient

In [1], Theorem 2.2, a necessary and sufficient condition is given for a subset of an inverse semigroup to generate a free inverse subsemigroup. However one very obvious further condition is omitted. The result should read as follows.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

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