TY - JOUR

T1 - The ideal structure of nilpotent-generated transformation semigroups

AU - Marques-Smith, M.P.O.

AU - Sullivan, Robert

PY - 1999

Y1 - 1999

N2 - In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

AB - In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

U2 - 10.1017/S0004972700036418

DO - 10.1017/S0004972700036418

M3 - Article

SN - 0004-9727

VL - 60

SP - 303

EP - 318

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

ER -