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A study of universal algebras in fuzzy set theory

**Authors:**Murali, V**Date:**1988**Subjects:**Fuzzy sets Algebra, Universal**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5394 , http://hdl.handle.net/10962/d1001983**Description:**This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting**Full Text:****Date Issued:**1988

**Authors:**Murali, V**Date:**1988**Subjects:**Fuzzy sets Algebra, Universal**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5394 , http://hdl.handle.net/10962/d1001983**Description:**This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting**Full Text:****Date Issued:**1988

Abstract Algebra: MAT 311

**Authors:**Makamba, B B , Murali, V**Date:**2011-06**Language:**English**Type:**Examination paper**Identifier:**vital:17606 , http://hdl.handle.net/10353/d1009981**Description:**Algebra: MAT 311, degree examination June 2011.**Full Text:**false**Date Issued:**2011-06

**Authors:**Makamba, B B , Murali, V**Date:**2011-06**Language:**English**Type:**Examination paper**Identifier:**vital:17606 , http://hdl.handle.net/10353/d1009981**Description:**Algebra: MAT 311, degree examination June 2011.**Full Text:**false**Date Issued:**2011-06

Complex Analysis: MAT 322

History and Fundamental Concepts of Mathematics: MAT 304

**Authors:**Van Dyk, T J , Murali, V**Date:**2010-01**Language:**English**Type:**Examination paper**Identifier:**vital:17592 , http://hdl.handle.net/10353/d1009966**Description:**History and Fundamental Concepts of Mathematics: MAT 304, Supplementary examination January/February 2010.**Full Text:**false**Date Issued:**2010-01

**Authors:**Van Dyk, T J , Murali, V**Date:**2010-01**Language:**English**Type:**Examination paper**Identifier:**vital:17592 , http://hdl.handle.net/10353/d1009966**Description:**History and Fundamental Concepts of Mathematics: MAT 304, Supplementary examination January/February 2010.**Full Text:**false**Date Issued:**2010-01

Measure and Integration Theory (Mathematics Honours): MAT 508

**Authors:**Makamba, B B , Murali, V**Date:**2010-01**Language:**English**Type:**Examination paper**Identifier:**vital:17594 , http://hdl.handle.net/10353/d1009968**Description:**Measure and Integration Theory (Mathematics Honours): MAT 508, degree examination January 2010.**Full Text:**false**Date Issued:**2010-01

**Authors:**Makamba, B B , Murali, V**Date:**2010-01**Language:**English**Type:**Examination paper**Identifier:**vital:17594 , http://hdl.handle.net/10353/d1009968**Description:**Measure and Integration Theory (Mathematics Honours): MAT 508, degree examination January 2010.**Full Text:**false**Date Issued:**2010-01

Preferential fuzzy sets: a key to voting patterns

**Authors:**Murali, V**Date:**2008-04-15**Language:**English**Type:**Text**Identifier:**vital:579 , http://hdl.handle.net/10962/d1012406**Description:**There is a one-to-one correspondence between ordered partitions and kernels of fuzzy subsets under a natural equivalence relation on them called preferential equality, on any n-element set Xn. We discuss some aspects of this correspondence with respect to counting voter’s choice or preference through the notions of Flags, Keychains and Pinned-flags.**Full Text:****Date Issued:**2008-04-15

**Authors:**Murali, V**Date:**2008-04-15**Language:**English**Type:**Text**Identifier:**vital:579 , http://hdl.handle.net/10962/d1012406**Description:**There is a one-to-one correspondence between ordered partitions and kernels of fuzzy subsets under a natural equivalence relation on them called preferential equality, on any n-element set Xn. We discuss some aspects of this correspondence with respect to counting voter’s choice or preference through the notions of Flags, Keychains and Pinned-flags.**Full Text:****Date Issued:**2008-04-15

Universal Algebra (Mathematics Masters): MAT 702

**Authors:**Makamba, B B , Murali, V**Date:**2010-01**Language:**English**Type:**Examination paper**Identifier:**vital:17595 , http://hdl.handle.net/10353/d1009969**Description:**Universal Algebra (Mathematics Masters): MAT 702, degree examination January 2010.**Full Text:**false**Date Issued:**2010-01

**Authors:**Makamba, B B , Murali, V**Date:**2010-01**Language:**English**Type:**Examination paper**Identifier:**vital:17595 , http://hdl.handle.net/10353/d1009969**Description:**Universal Algebra (Mathematics Masters): MAT 702, degree examination January 2010.**Full Text:**false**Date Issued:**2010-01

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