Representations of semigroups and the translational hull of a regular Rees matrix semigroup

Petrich, Mario

doi:10.1090/s0002-9947-1969-0246984-3

Cited by 10 publications

(5 citation statements)

References 13 publications

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“…These results go somewhat further than the previously published studies of multiplicative behavior of H, a subject whose importance for semigroup theory is apparent from the repeated use throughout the theory of the Miller-Clifford theorem, or from the arisal out of one key multiplicative lemma of the Petrich representation [12] and the Schiitzenberger representation (see [6]). Although the general behavior is significantly more complex, the author feels his results should have been obtained some years ago.…”

Section: = G • Y With G G G(k) H G G*(h) Then Ax = By If and Onlymentioning

confidence: 63%

The multiplicative behavior of ℋ

Grillet

1976

Trans. Amer. Math. Soc.

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ABSTRACT. Various results are given describing the product of two H-classes in an arbitrary semigroup in terms of groups and homomorphisms.1. It is known that in an arbitrary semigroup S the relation H of Green need not be a congruence: the product HK of two tf-classes H and K is usually spread out between several H-classes. Our main results give a fairly precise description of the set HK and the multiplication H x K -*■ HK. This is achieved by the first four of our five main theorems. Let {Rj)¡e¡, 0\)\ga he the families of all R-and L-classes which intersect HK. The first main theorem gives a partition of H into sets {Bj)iBI (and a partition of K into sets (YX

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Section: = G • Y With G G G(k) H G G*(h) Then Ax = By If and Onlymentioning

confidence: 63%

The multiplicative behavior of ℋ

Grillet

1976

Trans. Amer. Math. Soc.

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“…This construction of every completely regular semigroup from a semilattice, a family of Rees matrix semigroups and families of functions among their ingredients is refined in two further papers [3,43], the former written jointly with Clifford (see also [14,Theorem VI.5.2]). In [29] Mario provides a fundamental faithful representation, called the Petrich representation in [14, Theorem VI.2.3], for every regular semigroup S by means of translational hulls of completely 0-simple and completely simple semigroups. The completely 0-simple and simple semigroups involved are the traces of the D-classes of S. Let us call attention also to [42] where the main tool is the translational hull of rings and, among others, a new access to Everett sums (these correspond to the Schreier extensions of groups) is presented.…”

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confidence: 99%

A tribute to Mario Petrich

Szendrei

2021

Semigroup Forum

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“…This construction of every completely regular semigroup from a semilattice, a family of Rees matrix semigroups and families of functions among their ingredients is refined in two further papers [3] and [43], the former written jointly with Clifford (see also [14,Theorem VI.5.2]). In [29] Mario provides a fundamental faithful representation, called the Petrich representation in [14, Theorem VI.2.3], for every regular semigroup S by means of translational hulls of completely 0-simple and completely simple semigroups. The completely 0-simple and simple semigroups involved are the traces of the D-classes of S. Let us call attention also to [42] where the main tool is the translational hull of rings and, among others, a new access to Everett sums (these correspond to the Schreier extensions of groups) is presented.…”

mentioning

confidence: 99%

A tribute to Mario Petrich

Szendrei¹

2021

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has had a great influence on semigroup theory and on several generations of semigroup theorists. His extensive and deep contributions to the theory of regular, in particular, completely regular and inverse semigroups, and to a great variety of other subjects, earned him a special place in the history of our relatively young area. He is appreciated and admired by many of the people who have got the chance to know him and have benefitted from his insight and ideas.Mario Petrich was born in 1932 in Split, at that time in Yugoslavia. He spent part of his childhood and adolescence there under Italian rule, during World War II. Then Split was recovered by Yugoslavia, and he lived there until he was 20 years old, completing three years of studies in mathematics at the University of Zagreb. Then he left Yugoslavia, first went to West Germany and then to the USA. There he first worked as a messenger boy in New York at $1 per hour for 7 months, and then continued his studies. He completed his M.Sc. at Midwest University, and took his Ph.D. at the University of Washington, Seattle in 1961. After working at various institutions for shorter terms, partly in applied mathematics, he took a position at Pennsylvania State University in 1964. In 1976 he left the United States for personal reasons. He became a 'citizen of the world', accepting limited term appointments. Over the years, he was hosted, as a visiting professor or such, by many universities in various countries where he engaged in many successful collaborations: in

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Representations of semigroups and the translational hull of a regular Rees matrix semigroup

Cited by 10 publications

References 13 publications

The multiplicative behavior of ℋ

The multiplicative behavior of ℋ

A tribute to Mario Petrich

A tribute to Mario Petrich

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