A Study On Falling Fuzzy Theories Of EQ-algebras And States On Pseudo-BCI Algebras

EQ-algebra is a kind of algebra of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). BCI-algebra is an algebraic formulations of BCI-systems in the combinatory logic, and pseudo-BCI algebras is an extension of BCI-algebra. As is well known that the falling shadow representation theory directly relates probability concepts to the membership function of fuzzy sets and the state is a generalized probability measure, the paper aims to investigate the prefilter theory of EQ-algebras based on the falling shadow theory and study the states theory on local bounded pseudo-BCI algebras. The main research works are as follow:1. Studying the falling shadow theory on EQ-algebras by combining fuzzy sets and probability theories.First, the notions of falling fuzzy prefilters of EQ-algebras are introduced, and the relationships between falling fuzzy prefilters and fuzzy prefilters are provided. The notion of falling fuzzy prefilter is a generalization of that of fuzzy prefilter. Some of their characterizations are also presented.Second, the notions of falling fuzzy positive implicative (implicative, fantas-tic) prefilters are also proposed. They are proved to be generalizations of those of fuzzy positive implicative (implicative, fantastic) prefilters. Some examples are provided to show the existence of the three types of falling fuzzy prefilters. More-over some of their characterizations are displayed.Finally, the relationships among these special falling fuzzy prefilters are mainly investigated by using their characterizations. The main results are as fol- low:(1) Every falling fuzzy implicative prefilter with the weak exchange principle is a falling fuzzy positive implicative prefilter, but the converse is not true in gen-eral. (2) Every falling fuzzy implicative prefilter with the weak exchange principle is a falling fuzzy fantastic prefilter, but the converse is not true in general. (3) Every falling fuzzy prefilter is falling fuzzy implicative prefilter if and only if it is both a falling fuzzy positive implicative prefilter and a falling fuzzy fantastic prefilter.2. Studying the state theories on pseudo-BCI algebras by local bounded meth-ods.First, by introducing the notion of pseudo-atoms of which the branches are composed, we discuss the structure of pseudo-BCI algebras and get that any pseudo-BCI algebra is a union of it’s branches. Also we propose the notion of local bounded pseudo-BCI algebras and study some related properties,which en-sure the study on theories of states on pseudo-BCI algebra.Second, we define a Bosbach state on a local bounded pseudo-BCI algebras and discuss some basic properties about it, which extend the theories of states on pseudo-BCI algebras. Based on this, we study the quotient structure of pseudo-BCI algebras. If s is a Bosbach state of a local bounded pseudo-BCI algebra A, we prove that Afker(s) is equivalent to an MV-algebra. We also introduce the notion of state-morphisms on local bounded pseudo-BCI algebras and discuss the rela-tions between Bosbach states and state-morphisms. We provide that the extremal Bosbach states on local pseudo-BCI algebras are equivalent to state-morphisms.Finally, by discussing the existence of Bosbach states on weak semisimple lbp-BCI algebras, we study the existence of Bosbach states on local bounded pseudo-BCI algebra. We get that any associative lbp-BCI algebra does not admit a Bosbach state.