On A Semilattice Of Inverse Semigroups

In this dissertation, we characterize the congruences and congruence pairs on a semi-lattice of inverse semigroups by the congruences and congruence pairs on those inverse semigroups, and construct an isomorphism between the lattice , a family of all the normal congruence pairs on a semilattice of inverse monoids, and the lattice B, all the standard congruence pairs on the semilattice of inverse monoids. We discuss the relations between a natural partial order on a semilattice of inverse semigroups and that on every inverse semigroup. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we characterize the congruence pairs on a semilattice of inverse monoids by the congruences pairs on those inverse monoids and construct an isomorphism between the lattice A, a family of all the normal congruence pairs on a semilattice of inverse monoids, and the lattice B, all the standard congruence pairs on the semilattice of inverse monoids. Finally, we prove that a sublattice of the direct product of the lattices of congruences on those inverse monoids is isomorphic to a sublattice of the lattice of congruences on the semilattice of inverse monoids.Theorem 2.1.10 Let 5 be a semilattice of inverse monoids, and a family of all the normal congruence pairs of 5,(N, T) is defined by :Then are both complete lattices, moreover, there is a lattice isomorphism between A and B.Theorem 2.2.6 Let 5 = (Y; Sa) be a semilattice of inverse monoids SQ , for every where p is the congruence on S induced by y. Then(f is an isomorphism from C, the lattice of admissible congruences on the semilattice of 5Q, to C\, the sublattice of congruences on S.In Chapter 3, Firstly, we discuss a natural partial order on Sa and on a semilattice of inverse semigroups 5a, we obtain that a semilattice of inverse monoids Sa is a subdirect product SQ with a zero adjoined possibly. Secondly, we adjoin an identity to an inverse semigroup and make it an inverse monoid , and get a necessary and sufficient condition for a quotient semigroup of a semilattice of inverse semigroups to be a semilattice of quotient inverse semigroups. Finally, we characterize the group congruence on a semilattice of inverse semigroups by the group congruence on each inverse semigroup.Theorem 3.3.2 Let e a semilattice of inverse monoids is a semilattice congruence on Sa. If is a family of admissible congruences of 5, then p induced by is a semilattice congruence on 5, and p = 6, if 770 is a least semilattice congruence on is a least semilattice congruence on S, andJ is a semilattice Y In Chapter 4, we characterize congruence pairs and congruences on a semilattice of inverse semigroups Sby those inverse semigroups, without identities, and characterize group congruence on a semilattice of inverse semigroups by group congruences on those inverse semigroups.Theorem 4.2.4 Let S = (Y; be a semilattice of inverse semigroups is a family of left normal admissible congruences of S , a relation p on S is defined by...