Some new operations and the reticulation on L-algebras

Xiao Long Xin

Xian International University, Xian Polytechnic University

xlxin@nwu.edu.cn

Abstract：The reticulation of an algebraic structure A is a bounded lattice L(A) whose prime spectrum of ideals (or filters) is homeomorphic to the prime spectrum of ideals (or filters) of A. In this talk, we will induce some new operations on L-algebras, which come from the equivalent characterizations of the corresponding operations in MV-algebras.  We discuss the properties of the induced operations on bounded L-algebras, bounded K-algebras and bounded CKL-algebras, respectively. Then, we use the new operations "" and "" to propose the axiomatic system of reticulation on L-algebras. By use of the reticulation, we establish the axiom of reticulation on L-algebras, and finally prove that on a CKL-algebra A with join and (L,) of the reticulation of A, the spectrum of ideals of A is homeomorphic to the spectrum of filters of L. Finally, we discuss the existence of pre-reticulations on L-algebras. We give a characterization theorem of the existence of pre-reticulations on L-algebras, that is, on a CKL-algebra A,  A admits a pre-reticulation (L,λ) iff  A has a strong prime ideal. We give some examples to show how we can determine that an L-algebra A has a pre-reticulation and reticulation by the characterization theorem.