Regular Residuated Lattice * - Ideal And Its Properties

In the study of logic inference systems and logic algebra systems, ideals and filters are two important concepts and tools. The purpose of this paper is to introduce the *-operation and study ideals related to the *-operation using algebraic tools and topological techniques on regular residuated lattices. This study can establish links among logic, algebra and topology.Chapter I of this paper introduces basic concepts and results of partial order, implication operators, topology and ideals as preliminaries.Chapter II defines the *-operation in residuated lattices and investigates properties of regular residuated lattices and their *-operation. We also explore further relationships between regular residuated lattices with *-operation and other logic algebras, such as BCK algebras, lattice implication algebras, pre-linear residuated lattices. It is proved that regular residuated lattices with *-operation are all bounded BCK algebras. Some sufficient conditions for regular residuated lattices with *-operation to be implication algebras or pre-linear residuated lattices are obtained.Chapter III defines *-ideals and quasi *-ideal in regular residuated lattice with *-operation and investigates their properties and relationships. It is proved that *- ideal is quasi *-ideal. An example is given to show that the converse of the theorem is not true. A comprehensive characterization theorems is obtained for *-ideals: a non-empty subset I of a regular residuated lattice L with *-operation is a *-ideal if and only if one of the following five groups of conditions holds:(1) (?)a, b, c∈L, (i) 0∈I (ii) if a*b∈I and b*c∈I then a*c∈I;(2) (?)a, b, c∈L, (i) 0∈I (ii) if a*b∈I and b(?)c∈I then a(?)c∈I;(3) (?)a, b, c∈L, (i) 0∈I (ii) if b*a∈I and câ†'a∈I then câ†'b∈I;(4) (?)a, b, c∈L, (i) 0∈I (ii) if b*a∈I and bâ†'c∈I then aâ†'c∈I;(5) (?)a, b, c∈L, (i) 0∈I (ii) if aⁿ*b∈I and (b*c)*(?)a^m∈I then a^m+n*c∈I, where a²=a(?)a, a^m+n=a^m(?)aⁿ, a⁰=1.Finally, Chapter IV discusses congruence relations and prime *-ideals in regular residuated lattices with *-operation and investigates properties of prime *-ideal. We introduce a topology T, called the prime *- ideal topology, on the set *pI (L) of all prime *-ideals of a regular residuated lattice with certain conditions. It is proved that topological space (*pI (L), T) is a compact T₀-space. A necessary and sufficient condition is also given for topological space (*pI (L) , T) to be a T₁-space.