Fundamental Inverse Semigroups

Munn, W. D.

doi:10.1093/qmath/21.2.157

Cited by 123 publications

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“…A word in & Y is said to be reduced if it does not contain a syllable xx' 1 or a syllable x 444 G. B. Preston [2] If g is an element of G we shall denote its inverse in G by g~1. Set and set F 1 = F u {1}.…”

Section: Introductionmentioning

confidence: 99%

“…In this note we give an alternative description of free inverse semigroups. What Scheiblich did was to construct a free inverse semigroup as a semigroup of isomorphisms between principal ideals of a semilattice E, say, thus realising free inverse semigroups as inverse subsemigroups of the semigroup T E , a kind of inverse semigroup introduced and exploited by W. D. Munn [2]. We go instead directly to canonical forms for the elements of a free inverse semigroup.…”

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confidence: 99%

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Free inverse semigroups

Preston

1973

J. Aust. Math. Soc.

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Communicated by M. F. NewmanIn an important recent paper H. E. Scheiblich gave a construction of free inverse semigroups that throws considerable light on their structure [1]. In this note we give an alternative description of free inverse semigroups. What Scheiblich did was to construct a free inverse semigroup as a semigroup of isomorphisms between principal ideals of a semilattice E, say, thus realising free inverse semigroups as inverse subsemigroups of the semigroup T E , a kind of inverse semigroup introduced and exploited by W. D. Munn [2]. We go instead directly to canonical forms for the elements of a free inverse semigroup. The connexion between our construction and that of Scheiblich's will be clear. There are several alternative procedures possible to reach our construction on which we comment on the way.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Free inverse semigroups

Preston

1973

J. Aust. Math. Soc.

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“…A useful description of Green's relations on any inverse subsemigroup of the symmetric inverse semigroup on a set was given by Munn [10]. Since 2P$i(S)^y(S) for an inverse semigroup S, we can apply Munn's result to RESULT …”

Section: Restricting the Ideal Structure Of 0>d{s)mentioning

confidence: 99%

Inverse semigroups with certain types of partial automorphism monoids

Goberstein

1990

Glasgow Math. J.

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0. Abstract. For an inverse semigroup 5, the set of all isomorphisms between inverse subsemigroups of 5 is an inverse monoid under composition which is denoted by 2P$4(S) and called the partial automorphism monoid of 5. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called "exceptional" groups) all inverse semigroups 5 such that 0>$l(S) is completely semisimple. Secondly, for an inverse semigroup S, we find a convenient description of the greatest idempotent-separating congruence on SPst(S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental. Introduction.Let 5 be an inverse semigroup. For any subset X of 5, let (X) denote the inverse subsemigroup generated by X. To express the fact that a subset W c 5 is an inverse subsemigroup of 5, we write H =s 5. It will be assumed that 0 =s S. The semilattice of idempotents of 5 will be denoted by E s and the order of an element x e S by o(x). A partial automorphism of 5 is any isomorphism between inverse subsemigroups of S. The set of all partial automorphisms of 5 is denoted by 0>s4(S). Note that 0 e 9>M{S) since 0 can be considered as an isomorphism of the empty inverse subsemigroup of S onto itself. It is easy to see that with respect to composition 0>si(S) is an inverse semigroup; moreover, it is an inverse subsemigroup of S{S), the symmetric inverse semigroup on the set 5. In particular, the idempotents of 9>sd{S) are precisely the identity mappings i H for every H =£ S. Clearly i B ( = 0) is the zero and t s the identity of &s£(S). Thus 3>si(S) is an inverse monoid with zero; it is called the partial automorphism monoid of S. The group of units of &s$(S) is the automorphism group of 5. Since i w o i* = i H nK for an y H, K^S, the semilattice of idempotents of ^jrf(S) is, in fact, a lattice isomorphic to the lattice of all inverse subsemigroups of S.In [3] the author studied conditions under which an inverse semigroup 5 is determined by &si(S). Here we will investigate how certain natural restrictions imposed on SPst^) reflect upon the structure of 5. It should be noted that a similar problem for lattices of substructures of mathematical structures such as groups, rings, semigroups, inverse semigroups, etc., was the topic of numerous publications (see, for example, [14, Chapter I], [12, Chapter V] or [13, Chapter III]). However relatively few results on the above-mentioned problem for partial automorphism monoids have been established (see [4] for a brief survey). In fact, the author is aware of only one result which describes how the structure of an inverse semigroup S is influenced by imposing a certain restriction on &d{S): the determination by Kirkwood [7] of the structure of all groups whose partial automorphism ...

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“…Since 3 X is finite it has a principal series and is completely semisimple. If a is an idempotent its ^-class is {βGi x ; D(β) = R(β) = D(a)} (see [4]), which is the symmetric group on D(a). So for some a a non group principal factor has Δ-basic group isomorphic to P D{a) .…”

Section: Let S Be a Completely Semisimple Inverse Semigroup With Prinmentioning

confidence: 99%

Completely semisimple inverse Δ-semigroups admitting principal series

Trotter¹,

Tamura²

1977

Pacific J. Math.

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A Δ-semigroup is a semigroup whose lattice of congruences is a chain with respect to inclusion. A completely semisimple inverse Δ-semigroup that admits a principal series is characterized here as a semigroup that results from a particular series of ideal extensions of Brandt semigroups by Brandt semigroups. A characterization is given of finite inverse Δ-semigroups in terms of groups, Brandt semigroups, and one to one partial transformations of sets. Introduction.A A-semigroup is a semigroup whose lattice of congruences is a chain with respect to inclusion. Schein [8] and Tamura [ 11] showed that a commutative Δ-semigroup is either a quasicyclic group A, or a commutative nil semigroup B with the divisibility chain condition, or A 0 , or B\ We study here the structure of completely semisimple inverse Δ-semigroups with principal series. Such semigroups will be characterized in terms of Δ-groups, idempotent properties, and ideal extensions of Brandt semigroups by Brandt semigroups.In [11] it was shown that the least semilattice congruence on a Δ-semigroup has at most two classes. We begin by characterizing completely semisimple inverse semigroups admitting principal series and having this property.In the final section we show that each finite inverse Δ-semigroup determines a set of structure data that involves groups, Brandt semigroups and one to one partial transformations of sets. Conversely the semigroup can be reconstructed from the structure data.

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Fundamental Inverse Semigroups

Cited by 123 publications

References 0 publications

Free inverse semigroups

Free inverse semigroups

Inverse semigroups with certain types of partial automorphism monoids

Completely semisimple inverse Δ-semigroups admitting principal series

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