On Studies Of Archimedean Ordered Hypersemigroups

This article considers some problems of archimedean ordered hypersemigroups.To serve as a prelude,we give the concepts of strongly archimedean ordered hypersemigroups and strongly nil extension properties on an ordered hypersemigroup.Accordingly,the necessary definition of the generation and the fundamental of an ordered hypersemigroup are firstly given.Based on the concept of the generation of an ordered hypersemigroups,some results under the condition of included,anti-included,linked,well-combined ordered hypersemigroups are promoted in a natural manner.In order to investigate the property of the hyperideal in an ordered hypersemigroup,the description of hyperideal generation would be obtained.The specific layout is as follows:In the first chapter,we come to some certain background of the investigation in semigroups,ordered semigroups and hyperalgebra structures.Furthermore,we put forward a summary of archimedean semigroups as well as archimedean ordered semigroups,and introduce the prepared knowledge of this article.In the second chapter,the concrete construction of the quotient ordered semigroups by Rees congruences in an ordered semigroup is extended to that in an ordered hypersemigroup,and therefore we draw the conclusion that an ordered hypersemigroup is strongly nil extension of a simple ordered hypersemigroup if and only if it is strongly archimedean and contains a nonempty intra-regular subset.In the third chapter,we introduce the concept of C(S)of an ordered hypersemigroup,thus seeking out the least strong congruence in an ordered hypersemigroup.Naturally,the definition of included,anti-included,linked,well-combined ordered hypersemigroups and their equivalent description along with the construction of C(S)are given.The main results of this chapter are semilattices of archimedean ordered subhypersemigorups in a well-combined ordered hypesemigroup and bands of weakly r-archimedean ordered hypersemgroup in a right well-combined ordered hypersemigroup.In the fourth chapter,the notion of the generation and fundamental of an ordered hypersemigroup is given.On this basis,we introduce the idea of hyperideal generation and its fundamental by representing the maximal hyperideal generation and minimal hyperideal generation in which the fundamental is fixed.In this chapter,the characterization of simple and 0-simple ordered hypersemigroups,and the property of 0-archimedean ordered hypersemigroups with regard to hyperideal generation are laid out.In the fifth chapter,we derive some other results in a well-combined ordered hypersemigroups,which are,properties of left weakly archimedean ordered hypersemigroup,proper archimedean hyperideals and radicals in an ordered hypersemigroup.