On Congruences On Strong Semilattice Of Semigroups

In this dissertation, we characterize the congruences on a strong semilattice of semigroups by the congruences on those semigroups and prove that a sublattice of the direct product of the lattices of congruences on those semigroups is isomorphic to a sublattice of the lattice of congruences on the strong semilattice of semigroups. Besides the study of general semigroup, the strong semilattice of inverse semigroups, bands, and normal bands are discussed. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we characterize the congruences on a strong semilattice of semigroups by the congruences on those semigroups and prove that a sublattice of the direct product of the lattices of congruences on those semigroups is isomorphic to a sublattice of the lattice of congruences on the strong semilattice of semigroups. Finally, we get a necessary and sufficient condition for a quotient semigroup of a strong semilattice of semigroups to be a strong semilattice of quotient semigroups.Theorem 2.1.4 Let S = [Y; Sa; a,b], Pa is the congruence on is a class of admissible congruences. A relation p on S is defined by:Then p is a congruence on S, and for all Theorem 2.2.5 Let S - [Y;Sa; a,B], and each is an epimorphism. Define a map where p is the congruence on S induced by Then is an isomorphism from C, the lattice of admissible congruences on the strong semilattice of Sa, to , the sublattice of congruences on S.In Chapter 3 , we characterize the congruences on a strong semilattice of inverse semigroups by the congruence on each inverse semigroup.Theorem 3.1.6 Let S = [Y; Sa; a,B] be a strong semilattice of inverse semigroups and (N, T) be a congruence pair of S. For all a in Y, denote and and Ta, respectively. Then is a congruence pairs of Sa. Furtherly, if N and T satisfy:(iii) If a 6 Sa, then (iv) , then (v) {r |#a }aey is a class of admissible congruences, Then (JV, r) is exactly the congruence pair of 5 induced by Theorem 3.1.7 Let be a strong semilattice of inverse semigroups. If pa is a congruence on satisfy (2.1.1), satisfy (3) and (4), then the relation p on S defined by(a, b) is a congruence on 5 induced by Conversely, if p is a congruence on 5, then for every a in Y, p |sa is a congruenceon Sa. Furtherly, if p satisfies: Kerp andthen p is exactly the congruence on 5 induced by p|Sa.In chapter 4 , we mainly discuss the least semilattice congruence on strong semilattice of bands.Theorem 4.2 Let B = be the strong semilattice of bands Ba(a Y),a be the least semilattice congruence on Ba. Thenis the least semilattice congruence on B, and B/ is the strong semilattice of In chapter 5 , we mainly discuss the least inverse semigroup congruence on a strong semilattice of orthodox semigroups.Theorem 5.5 Let be the strong semilattice of orthodox semigroups Sa. If ra is the least inverse semigroup congruence on 5s, , and a relation on S is defined by , i.e.that for a, b S,then 7 is the least inverse semigroup congruence on S, and Conversely , if 7 is the least inverse semigroup congruence on 5, then is the least inverse semigroup congruence on Sa, and...