Strongly (Ordered) Regular Equivalence Relations On (Ordered) Semihypergroups

Algebraic hyperstructure theory was first introduced by Marty in 1934. In the 1980s and 1990s, Kepka, Jezec, Nemec et al. studied semihypergroups on the background of semigroup theory. Algebraic hyperstructures are a generalization of classical algebraic structures. Semihypergroups are the simplest algebraic hyperstructures which are a gen-eralization of the concept of semigroups. As we all know, congruence relations play an important role in studying semigroups. Similarly, (strongly) regular equivalence relations have effects on semihypergroups. A regular equivalence relation and a strongly regular equivalence relation on a semihypergroup are respectively such that the corresponding structures are a semihypergroup and a semigroup. The least strongly regular equivalence relation on a semihypergroup was introduced by Koskas in 1970, called the fundamental relation and denoted by β*. The concept of a fuzzy set was introduced by Zadeh in 1965. In 1989, Nemitz introduced the concept of fuzzy relations on a set. Later on, the fuzzy congruence relations on semigroups were introduced and studied by Samhon. In 2000, Davvaz generalized the fuzzy relations to semihypergroups and introduced the concept of fuzzy strongly regular equivalence relations on semihypergroups. In 2008, Ameri ob-tained that both the set of all strongly regular equivalence relations and the set of all fuzzy strongly regular equivalence relations on a semihypergroup consist complete lattices. Or-dered semigroups can be obtained by integrations of semigroups and order relations. In 1993, Kehayopulu and her student Tsingelis introduced the concept of pseudoorders on ordered semigroups, which play the role in ordered semigroups similar to congruences on semigroups. Later on, Xiangyun Xie described clearly the congruences which are such that the quotient semigroups are ordered semigroups (non-trivial order) by applying the tool on ordered semigroups. In 2011, Heidari and Davvaz introduced the concept of or-dered semihypergroups by applying the integration to semihypergroups and then made deep studies. In 2015, Davvaz et al. introduced the concept of a pseudoorder on an or-dered semihypergroup and constructed a strongly regular equivalence relation for which the corresponding quotient structure is an ordered semigroup by applying it. However, they proposed the problem:is there a regular equivalence relation (non-strongly regular) on an ordered semihypergroup for which the corresponding quotient structure is an or-dered semihypergroup? Based on the former research and some problems on (ordered) semihypergroups, this thesis is mainly concerned with (fuzzy) strongly regular equiva-lence relations on semihypergroups and (strongly) ordered regular equivalence relations on ordered semihypergroups. The main work of this dissertation are as follows.In Chapter 1, we mainly introduce the research background, research developments, and achievements of (strongly) regular equivalence relations on semihypergroups and (strongly) ordered regular equivalence relations on ordered semihypergroups. In the end, we give a simple introduction of the main contents of this thesis.In Chapter 2, we study respectively the strongly regular relation generated by a binary relation and the fuzzy strongly regular equivalence relation generated by a fuzzy relation on a semihypergroup. As a consequence, the fundamental relation/β* and the least fuzzy strongly regular equivalence relation βf* on a semihypergroup are obtained. Furthermore, we respectively give descriptions of the greatest strongly regular equivalence relation contained in a given equivalence relation and the greatest fuzzy strongly regular equivalence relation less than a given fuzzy equivalence relation on a semihypergroup.In Chapter 3, we introduce the concept of (strongly) ordered regular equivalence relations on an ordered semihypergroup. Moreover, we construct an ordered semilattice equivalence relation on an ordered semihypergroup by applying hyperfilters. Finally, we construct an ordered regular equivalence relation on an ordered semihypergroup by hy-perideals, which is a positive answer to the open problem proposed by Davvaz et al. Meanwhile, the ordered regular equivalence relations on the direct product of semihyper-groups are studied.In Chapter 4, we give the fundamental normal homomorphism theorem of ordered semihypergroups. Moreover, we introduce the concept of p-chain and apply it to charac-terize the (strongly) ordered regular equivalence relations on an ordered semihypergroup. Finally, the subset of an ordered semihypergroup can serve as some ordered regular e-quivalence class is determined.