DIRECTLY INDECOMPOSABLE RESIDUATED LATTICES
Lavinia Corina Ciungu · Iranian journal of fuzzy systems · 2009
The aim of this paper is to extend results established by H. Ono and T. Kowalski regarding directly indecomposable commutative residuated lattices to the non-commutative case. The main theorem states that a residu- ated lattice A is directly indecomposable if and only if its Boolean center B(A) is {0,1}. We also prove that any linearly ordered residuated lattice and any local residuated lattice are directly indecomposable. We apply these results to prove some properties of the Boolean center of a residuated lattice and also define the algebras on subintervals of residuated lattices. It is known that the study of classical logic can be reduced to studying Boolean algebras. Therefore, the discussion of any type of non-classical logic raises a ques- tion about the corresponding abstract algebra. There has been much research in the field of fuzzy logic when the conjunction of the truth values structure is not necessarily commutative. Developing algebraic models for non-commutative multiple-valued logics is a central topic in the research of fuzzy systems and one such algebraic structure is the non-commutative residuated lattice. In the last few years a corresponding fuzzy theory has developed in parallel with the classical the- ory (2, 3, 19, 13). Commutative residuated lattices were first introduced by M. Ward and R.P. Dil- worth as a generalization of ideal lattices of rings. Recently, these structures have been studied in (10) and (18). Non-commutative residuated lattices, sometimes called pseudo-residuated lattices, biresiduated lattices or generalized residuated lat- tices, are the algebraic counterparts of substructural logics; i.e. logics which lack at least one of the three structural rules, namely contraction, weakening and exchange. Complete studies on non-commutative residuated lattices were developed in (1) and (17). The aim of this paper is to extend results proved by H. Ono and T. Kowalski for the case of commutative residuated lattices to the non-commutative case. The main theorem states that a residuated lattice A is directly indecomposable if and only if its Boolean center B(A) is {0,1}. We also prove that any local residuated lattice is directly indecomposable and derive some properties of the Boolean center of a residuated lattice. As an application of the Boolean center of a residuated lat- tice we prove that any subinterval (a,b) of a residuated lattice can be endowed with an algebraic structure of the same kind as the original one. These structures are