Structure of a permutable Munn semigroup of finite rank

Derech, V. D.

doi:10.1007/s11253-006-0107-4

Cited by 2 publications

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References 4 publications

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“…Using these remarks, we easily establish that the ideal I 1 is a Brandt semigroup. It was proved in [10] that any nonzero ideal of the semigroup T E is dense. Furthermore, it is clear that the structural group of the ideal I 1 is a singleton.…”

Section: Corollary 3 If σ Is a Stable Order On The Permutable Mann Smentioning

confidence: 99%

On maximal stable orders on an inverse semigroup of finite rank with zero

Derech

2008

Ukr Math J

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Section: Corollary 3 If σ Is a Stable Order On The Permutable Mann Smentioning

confidence: 99%

On maximal stable orders on an inverse semigroup of finite rank with zero

Derech

2008

Ukr Math J

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Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

Derech

2009

Ukr Math J

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We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental.The Munn semigroup (see [1]), i.e., the inverse semigroup of all isomorphisms between principal ideals of a semilattice with respect to the ordinary operation of composition of binary relations, plays a fundamental role in the theory of representations of inverse semigroups (see [2, p. 170]). This explains the importance of comprehensive investigation of these semigroups and their classification.In the present paper, we investigate the structure of a Munn semigroup of finite rank every order of which is fundamental or antifundamental (for definitions, see Sec. 1). The main result of this paper is Theorem 1. Main Definitions and TerminologyA semilattice E is called a semilattice of finite length if there exists a natural number n such that the length of any chain from E does not exceed n.Let E be a semilattice of finite length. It is obvious that it contains the least element, which we denote by 0. Let T E denote the Munn semigroup, i.e., the inverse semigroup of all isomorphisms between the principal ideals of the semilattice E with respect to the ordinary operation of composition of binary relations. It is clear that the transformation 0 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ is the least element of the semigroup T E . Denote the domain of definition and the range of values of the transformation f T E ∈ by dom ( ) f and im ( ) f , respectively. Let S be an arbitrary semigroup and let N 0 be the set of all nonnegative integers. A function rank : S → N 0 is called a rank function on the semigroup S if rank ( ) ab ≤ min ( ) rank a ( , rank ( ) b ) for any a b S , ∈ . The number rank ( )x is called the rank of the element x. Let S be an inverse semigroup whose lattice of idempotents has finite length. The function rank ( ) a = h aa -1 ( ) , where h aa -1 ( ) is the height of the idempotent aa -1 in the semilattice of idempotents of the semigroup S, is a rank function (see [3]). We say that an inverse semigroup is a semigroup of finite rank if the semilattice of its idempotents has finite length.Let f T E ∈ . If dom ( ) f = aE, then, by definition, rank ( ) f = rank ( ) a , where rank ( ) a is the height of the element a in the semilattice E. In [4], it was proved that rank : T E → N 0 is a rank function.An order relation τ on an arbitrary semigroup S is called fundamental (see [5] or [6, p. 289]) if the ordered semigroup ( ; ) S τ is O-isomorphic to a certain semigroup of partial transformations of a set ordered by in-

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Structure of a permutable Munn semigroup of finite rank

Abstract: A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank.

Cited by 2 publications

References 4 publications

On maximal stable orders on an inverse semigroup of finite rank with zero

On maximal stable orders on an inverse semigroup of finite rank with zero

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

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