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

Abstract

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-

Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental

“…Moreover, in Theorem 2, we characterize the semilattice of idempotents of the finite inverse semigroup with zero all stable orders of which are exhausted by fundamental and antifundamental. The paper continues and develops (for the finite case) the results established in [7,8]. The principal result of the present paper is formulated in Theorem 1.…”

“…We consider the Munn semigroup T E , i.e., the inverse semigroup of all isomorphisms of the principal ideals of the semilattice E relative to the ordinary operation of superposition of binary relations (see, e.g., [11]). Note that the semilattice of idempotents of the semigroup T E is isomorphic to the semilattice E. Thus, by virtue Theorem 1 from [8], every stable order on the inverse semigroup T E is either fundamental or antifundamental.…”

“…Thus, in [5], it was proved that any stable order on a symmetric inverse semigroup is fundamental or antifundamental. The structure of Munn semigroup of finite length each stable order of which is fundamental or antifundamental was clarified in [8]. It is clear that the problem of necessary and sufficient conditions for the stable orders on inverse semigroups to be exhausted by fundamental or antifundamental is quite urgent.…”