On simpleness of semirings and complete semirings

In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show that the concepts of simpleness, congruence-simpleness, and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; we also describe congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we obtain the following further results: The ideal-simpleness, congruence-simpleness, and simpleness of semirings are Morita invariant properties; a complete description of simple semirings containing the infinite element; the "Double Centralizer Property" representation theorem for simple semirings; a complete description of simple semirings containing a projective minimal one-sided ideal; a characterization of ideal-simple semirings having either an infinite element or a projective minimal one-sided ideal; settling a conjecture and a problem as published by Katsov in 2004 for the classes of simple semirings containing either an infinite element or a projective minimal left (right) ideal, showing, respectively, that semirings of those classes are not perfect and that the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.

The Application of Fuzzy Clustering Method in the Division of Reservoir Flow Unit

Based on the sedimentary character of six time cells in P1 Beierxi of Sabei developed region in Daqing oil field, this paper applies fuzzy mathematics method to divide reservoir flow unit. At present, there is no uniform division method, because flow unit is a fuzzy set with multi factor influence. There is great uncertainty among every influential factor, so does the influence extent of every factor to flow unit division. It isn’t objective to divide flow unit by one factor or simple congruence of several factors. However, the fuzzy mathematic method is fit to analyze the uncertainty relation in system or among systems. When the data of core well is in defect, applying the fuzzy clustering method to divide flow unit and combining development practice can reduce human factor greatly and make classification more objective and rational.

The natural partial order on a regular semigroup

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.