On a Class of Inverse Semigroups

Venkatesan, P. S.

doi:10.2307/2372863

Cited by 12 publications

(2 citation statements)

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“…A regular semigroup 5 is said to be h -k regular if S is h -k type. A generalization of P. S. Venkatesan's theorem [5] follows. THEOREM…”

Section: Vii) S Is Completely 0-simple Every O-minimal Right Ideal Rofs Is the Union Of A Right Group With Zero G° A Union Of H Disjoint mentioning

confidence: 89%

Completely 0-Simple and Homogeneous n Regular Semigroups

Kim¹

1966

Proceedings of the American Mathematical Society

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The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more that eight typewritten double spaced pages long will not be considered as acceptable. 1. In this note we state three new results (Theorems 1, 4 and 5) about the completely 0-simple and homogeneous n regular semi-groups. We follow the notation and terminology of [l] unless stated otherwise. Throughout, S denotes a semigroup with zero. Leta£S\0. Denote by V(a) the set of all inverses of a in S, that is, V(a) = (*£S: axa=a, xax-x). A semigroup S with zero is said to be homogeneous n regular if the cardinal number of the set V(a) of all inverses of a is n for every nonzero element a in 5, where n is a fixed positive integer. Let T be a subset of 5. We denote by E(T) the set of all idem-potents of 5 in T. 2. The next theorem is a generalization of R. McFadden and Hans Schneider's theorem [3]. THEOREM 1. Let S be a 0-simple semigroup and let n be a fixed positive integer. Then the following are equivalent. (i) S is a homogeneous n regular and completely 0-simple semigroup. (ii) For every a 9*0 in S there exist precisely n distinct nonzero elements (*•)?"! such that aXia=afor i = 1, 2, • • • , n and for all c, d in S cdc = C5*0 implies dcd = d. (iii) For every a 7*0 in S there exist precisely n distinct nonzero idem-potents (e i)i^ 1 = E a and k distinct nonzero idempotents (ƒ,)ƒ-1 = F a such that da=a = afj for i = l, 2, • • • , h,j = l, 2, • • • , fe, hk = n, E a contains every nonzero idempotent which is a left unit of a, F a contains every nonzero idempotent which is a right unit of a and E a C\F a contains at most one element. (iv) For every aj*0 in S there exist precisely k nonzero principal right ideals (2?»)Jli and h nonzero principal left ideals (£y)*»i such that 867

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“…A regular semigroup 5 is said to be h -k regular if S is h -k type. A generalization of P. S. Venkatesan's theorem [5] follows. THEOREM…”

Section: Vii) S Is Completely 0-simple Every O-minimal Right Ideal Rofs Is the Union Of A Right Group With Zero G° A Union Of H Disjoint mentioning

confidence: 89%

Completely 0-Simple and Homogeneous n Regular Semigroups

Kim¹

1966

Proceedings of the American Mathematical Society

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“…As a consequence of this theorem one obtains certain results concerning inverse semigroups with zero (cf. [5]). For in (a) and (b) one substitutes "regular" by "inverse" which is equivalent to uniqueness of e and ƒ in (c), and in (f) "completely 0-simple" is replaced by "Brandt semigroup"; the remaining items also have their analogues for this case.…”

Section: Let S Be a Semigroup With A Completely 0-simple Ideal M In mentioning

confidence: 99%

Some results concerning completely 0-simple semigroups

Lallement¹,

Petrich²

1964

Bull. Amer. Math. Soc.

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We follow the notation and terminology of [l ]. A semigroup T with zero is said to be O-rectangular if it has the property: if all the products at the vertices of a closed polygonal line (with a finite number of vertices) of the multiplication table are all but one equal to a nonzero element m and the remaining product is not zero, then it is also equal to m. A rectangular 0-band is a Rees matrix semigroup with zero over the one-element group. The semigroup in (f) need not be unique. We note that an analogous result is valid for completely simple semigroups (i.e., without zero) ; in such a case (b) and (c) remain essentially the same, (a) becomes S^GXE y E is a rectangular band, in (d) "O-rectangular" is replaced by "rectangular," in (e) it suffices to take four entries of P at a time, and (f) states that idempotents form a semigroup (and thus in this case the semigroup in (f) is unique).An ideal / of a semigroup T is said to be a matrix ideal of T if : for all a, b, cÇzT, (a) aTbQI implies a£7 or bÇil, (b) abc(£I implies abÇzI or bcÇzI.

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On decomposition of semigroups with zero

Venkatesan

1966

Math Z

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On a Class of Inverse Semigroups

Cited by 12 publications

References 1 publication

Completely 0-Simple and Homogeneous n Regular Semigroups

Completely 0-Simple and Homogeneous n Regular Semigroups

Some results concerning completely 0-simple semigroups

On decomposition of semigroups with zero

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