Advances in Adaptive Data Analysis Vol. 3, No. 3 (2011) 339–350 c  World Scientific Publishing Company DOI: 10.1142/S1793536911000714 MULTI-FUZZY EXTENSIONS OF FUNCTIONS SABU SEBASTIAN * and T. V. RAMAKRISHNAN Department of Mathematical Sciences, Kannur University, Kannur, Kerala-670567, India * sabukannur@gmail.com In this paper, we study various properties of multi-fuzzy extensions of crisp functions using order homomorphisms, complete lattice homomorphisms, L-fuzzy lattices, and strong L-fuzzy lattices as bridge functions. Keywords : Multi-fuzzy set; bridge function; multi-fuzzy extension. 1. Introduction We proposed the theory of multi-fuzzy sets [Sabu and Ramakrishnan (2010a), (2011a)] as an extension of theories of fuzzy sets, L-fuzzy sets [Goguen (1967)] and intuitionistic fuzzy sets [Atanassov (1986)]. Theory of multi-fuzzy sets deals with multi-level fuzziness and multi-dimensional fuzziness. Our previous papers dis- cussed the basic notions of multi-fuzzy sets, multi-fuzzy topology, and multi-fuzzy subgroups [Sabu and Ramakrishnan (2010a), (2010c), (2011b) (2011a)]. Bridge functions and multi-fuzzy extensions of crisp functions have great importance in the study of multi-fuzzy sets. In this paper, multi-fuzzy extensions of crisp func- tions based on the bridge functions such as order homomorphisms, complete lattice homomorphisms, L-fuzzy lattices, and strong L-fuzzy lattices are studied. 2. Preliminaries We use the following notations. X and Y stand for universal sets, I,J, and K stand for indexing sets, L and M stand for partially ordered sets, L X stands for the set of all functions from X to L, {L j : j ∈ J }, and {M i : i ∈ I } stand for families of complete lattices with order reversing involutions, unless it is stated otherwise. Partial order ≥ is the opposite order relation of the partial * Corresponding author. 339