FUZZY CLOSING OPERATORS AND THEIR COHERENCE AS FUZZY CONSEQUENCE OPERATORS J. Elorza 1 R. Fuentes-Gonz´ alez 2 J. Bragard 1 P. Burillo 2 1 Dept. Physics and Applied Mathematics. University of Navarra, {jelorza,jbragard}@unav.es 2 Dept. Automation and Computer Science. Public University of Navarra, {rfuentes,pburillo}@unavarra.es Abstract In a previous paper [7] we explored the no- tion of coherent fuzzy consequence operator. It is well-known that the operator induced by a fuzzy preorder through Zadeh’s compo- sitional rule is always a coherent fuzzy con- sequence operator. It is also known that the relation induced by a fuzzy consequence ope- rator is a fuzzy preorder if such operator is coherent [5]. Moreover fuzzy closing opera- tors of mathematical morphology can be con- sidered as fuzzy consequence operators. The aim of this paper is to analyze the coherence of such operators and also to consider their induced relations. Keywords: Fuzzy Preorder, Fuzzy Con- sequence Operator, Fuzzy Closing Opera- tor, Coherent Operators, Approximate Rea- soning, Mathematical Morphology, Image Processing. 1 INTRODUCTION Consequence Operators in classical logic were intro- duced by A. Tarski in 1930 [17]. According to Tarski, a logic is just a set of propositions with a consequence operator. The relationship between consequence ope- rators and preorders is well known [3]. Concepts of Fuzzy Preorder and Fuzzy Consequence Operator (FCO for short) are essential on fuzzy logic. These notions have been defined as a natural genera- lization of the classical ones: Given a non-empty universal set X which will repre- sent a set of propositions and a t-norm ∗, a fuzzy (bi- nary) relation R on X (fuzzy subset of X ×X) is called a fuzzy ∗-preorder if it satisfies: (R1) R(x, x)=1 ∀ x ∈ X (reflexivity) (R2) R(x, z) ≥ R(x, y) ∗ R(y,z) ∀ x, y, z ∈ X (∗- transitivity). If R is also symmetric, this is, R(x, y)= R(y,x) for all x, y ∈ X then it is called a fuzzy ∗-similarity, fuzzy ∗-indistinguishability or, equivalently, fuzzy ∗- equivalence. If R is only reflexive and symmetric, we will say that it is a fuzzy tolerance. Fix a complete lattice L which will be the range of the memberships of the fuzzy subsets of X, J. Pavelka introduced in 1979 the concept of FCO on X in fuzzy logic [14] extending the concept of consequence ope- rator in Tarski’s sense in a natural way. A function C : L X −→ L X is a FCO on X if it satisfies: (C1) μ ⊂ C(μ) for all fuzzy subset μ ∈ L X (inclusion) (C2) μ 1 ⊂ μ 2 = ⇒ C(μ 1 ) ⊂ C(μ 2 ) for all μ 1 , μ 2 ∈ L X (monotony) (C3) C(C(μ)) ⊂ C(μ) for all μ ∈ L X (idempotence) Notice that, under the inclusion axiom, (C3) may be written equivalently as (C3’) C(C(μ)) = C (μ) ∀ μ ∈ L X . These operators, called closure operators in the al- gebraic, logical and topological contexts, have been studied extensively [1],[2],[9],[19]. During the last and current decades, these operators have been also stu- died in the context of the fuzzy logic taking the chain L = [0, 1] as a special case [4],[5],[7],[8],[11]. In 1991, J.L. Castro and E. Trillas proved [5] the fo- llowing result: If R is a fuzzy ∗-preorder then the operator C ∗ R between fuzzy subsets of X given by the ESTYLF08, Cuencas Mineras (Mieres - Langreo), 17 - 19 de Septiembre de 2008 XIV Congreso Español sobre Tecnologías y Lógica fuzzy 221