Characterizations of Hemirings Based on Probability Spaces

Starting from a unified treatment of uncertainty by combining probability and fuzzy set theory [1], Goodman [2] put forward the equivalence between a fuzzy set and a class of random sets. Falling shadow representation theory was established based on the collection of Wang and Sanchez [3], which is directly related to the concept of probabilistic fuzzy set membership function. The theory shows the selection methods related to the joint degrees distributions. It provides a reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. Utilizing the theory of falling shadows, in particular, Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Yuan and Lee [6] considered a fuzzy subgroup (subring, ideal) as the falling shadow of a cloud of the subgroups (subrings, ideals). Jun and Kang [7] proposed a theoretical approach for BCK algebras. A semiring plays an important role in studying matrices and determinants. Many aspects of the theory of matrices and determinants over semiring have been studied by Beasley and Pullman [8] and Ghosh [9]. The ideals in semiring are useful for many purposes, but they do not coincide with the usual ring ideals if S is ring in general. Their use is thus somewhat limited in terms of obtaining analogues of ring theorems for semirings. In fact, many results in ring apparently have no analogues in hemirings using only ideals. LaTorre [10] investigated h-ideals and k-ideals in hemirings in an effort to obtain analogues of familiar ring theorems. The fuzzy theory in semirings and hemirings has been discussed by many researchers (see [11–16]). The concept of h-hemiregular hemirings has been introduced by Zhan and Dudek [17] to generalize the regularity in hemirings. Further, some characterizations of h-semisimple and h-intrahemiregular hemirings were investigated by Yin et al. [18, 19]. It is pointed out that some generalized fuzzy h-ideals of hemirings were investigated byMa et al.; for example, see [18–25]. Recently, some properties of falling fuzzy ideals of hemirings have also been investigated by Yu and Zhan [26]. As a continuation of our previous investigation of falling fuzzy ideals of hemirings, the present paper is organized as follows. In Section 2, we recall the concepts and properties of hemirings, fuzzy sets, and falling shadows. In Section 3, we introduce the concept of falling fuzzy h-ideals and investigated some related properties. Finally, we investigate characterizations ofh-hemiregular hemirings based on independent (prefect positive correlation) probability spaces in Section 4.