Completely Regular Congruence On The Semigroup And The Congruence Lattice,

The present dissertation is devoted exclusively to the theory of congruences on completely regular semigroups. It contains four chapters.In Chapter 1, certain basic definitions and results are presented. We state briefly most of the needed notion, terminology and preliminary. In §1.2, some basic concepts are covered: direct product, sub-direct product, spined product, and so forth. Then we prove some elementary results concerning these concepts. Finally, a construction of pure cover is given using the concept of "purity" introduced by M. Petrich.To each congruence p on a completely regular semigroup S, we may associate a possibly different congruence by applying to p the operator of forming lower T. For example, to a least regular band congruence, we obtain a least regular cryptogroup congruence. These results can be found in §2.1. In §2.2, we characterize orthodox congruences and give the sufficient and necessary conditions on which the kernel of an orthodox congruence is cryptic. Concerning the congruence pairs lattice CP(S) for some certain completely regular semigroups, in §2.3, we arrive at (sub)direct product of two complete lattice IC(S) which is the set of all normal subsets of S and TN{S) which is the set of all normal equivalence relations on E(S).In Chapter 3, we investigate certain relations on the congruence lattice C(S). We consider in §3.1 the relation on the congruence lattice of a completely regular semigroup S which identifies two congruences if they have the same intersection with Green relation C We also characterize congruences on a completely regular semigroup by means of so-called ζ-related congruences similarly to D-related congruences introduced by M. Petrich. In §3.2, the connection among relations K, T, L is analyzed. Finally, we classify all those semigroups S for which |C(5)/L| < 2.For any completely regular semigroup S, we denote by pL and pl (res. pG and pg) the greatest and the least congruence on S with the same local p ∩ D (res. global p V D) as p. Let Γ= {G,g, L, l}. Then the semigroup F(S) generated by the operators Γ on C(S) is called the GL-operator semigroup of S. In Chapter 4, firstly, we describe the GL-operator semigroup for general completely regular semigroups. Furthermore,...