Quotient MI-groups

A many identities group (MI-group, for short) is a special algebraic structure in which identity like elements (called pseudoidentities) are specified and collected into a monoidal substructure. In this way, many algebraic structures, such as monoids of fuzzy intervals (numbers) or convex bodies possessing behavior very similar to that of a group structure, may be well described and investigated using a new approach, which seems to be superfluous for the classical structures. The concept of MI-groups was recently introduced by Holcapek and St?pnicka in the paper "MI-algebras: A new framework for arithmetics of (extensional) fuzzy numbers" to demonstrate how a standard structure can be generalized in terms of MI-algebras. This paper is a continuation of the development of MI-group theory and is focused on the construction of quotient MI-groups and a specification of the conditions under which the isomorphism theorems for groups are fulfilled for MI-groups.

[1]  Svetoslav Markov,et al.  On the algebraic properties of convex bodies and some applications. , 2000 .

[2]  Milan Mares.  Addition of rational fuzzy quantities: Convolutive approach , 1989, Kybernetika.

[3]  Dug Hun Hong,et al.  Additive Decomposition of Fuzzy Quantities , 1996, Inf. Sci..

[4]  K. David Jamison,et al.  Possibilities as cumulative subjective probabilities and a norm on the space of congruence classes of fuzzy numbers motivated by an expected utility functional , 2000, Fuzzy Sets Syst..

[5]  Milan Mareš,et al.  Additive decomposition of fuzzy quantities with finite supports , 1992 .

[6]  Didier Dubois.  Fuzzy sets and systems , 1980 .

[7]  F. Jenik,et al.  Über die Nachrichtenverarbeitung in der Nervenzelle , 2004, Kybernetik.

[8]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .

[9]  George J. Klir,et al.  Constrained fuzzy arithmetic: Basic questions and some answers , 1998, Soft Comput..

[10]  Weldon A. Lodwick,et al.  Fundamentals of Interval Analysis and Linkages to Fuzzy Set Theory , 2008 .

[11]  A. Kaufmann,et al.  Introduction to fuzzy arithmetic : theory and applications , 1986 .

[12]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[13]  Svetoslav Markov.  On the Algebraic Properties of Intervals and Some Applications , 2001, Reliab. Comput..

[15]  Weldon A. Lodwick,et al.  Constrained Interval Arithmetic , 1999 .

[16]  Madan M. Gupta,et al.  Introduction to Fuzzy Arithmetic , 1991 .

[17]  Weldon A. Lodwick,et al.  Constrained intervals and interval spaces , 2013, Soft Comput..

[18]  Milan Mares,et al.  Weak arithmetics of fuzzy numbers , 1997, Fuzzy Sets Syst..

[19]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[20]  Didier Dubois,et al.  Gradual Numbers and Their Application to Fuzzy Interval Analysis , 2008, IEEE Transactions on Fuzzy Systems.

[21]  Milan Mares,et al.  Addition of fuzzy quantities: Disjunction-conjunction approach , 1989, Kybernetika.

[22]  Azriel Rosenfeld,et al.  Fuzzy Group Theory , 2005, Studies in Fuzziness and Soft Computing.

[23]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[24]  Dong Qiu,et al.  Symmetric fuzzy numbers and additive equivalence of fuzzy numbers , 2013, Soft Computing.

[25]  Wei Zhang,et al.  Algebraic properties and topological properties of the quotient space of fuzzy numbers based on Mareš equivalence relation , 2014, Fuzzy Sets Syst..

[26]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[27]  Svetoslav Markov,et al.  On quasilinear spaces of convex bodies and intervals , 2004 .

[28]  R. Tennant.  Algebra , 1941, Nature.

[29]  John N. Mordeson,et al.  Fuzzy commutative algebra , 1998 .

[31]  Radko Mesiar,et al.  Fuzzy Interval Analysis , 2000 .

[32]  Martin Stepnicka,et al.  On Isomorphism Theorems for MI-groups , 2013, EUSFLAT Conf..

[33]  George J. Klir,et al.  Fuzzy arithmetic with requisite constraints , 1997, Fuzzy Sets Syst..

[34]  Alexandru Mihai Bica.  Algebraic structures for fuzzy numbers from categorial point of view , 2007, Soft Comput..

[35]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[36]  Mirko Navara,et al.  How to make constrained fuzzy arithmetic efficient , 2001, Soft Comput..

[37]  Didier Dubois,et al.  Gradual elements in a fuzzy set , 2008, Soft Comput..

[38]  Weldon A. Lodwick,et al.  Interval and Fuzzy Analysis: A Unified Approach , 2007 .

[39]  M. Mares,et al.  Computation Over Fuzzy Quantities , 1994 .

[40]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

[42]  A. Bica.  Categories and algebraical structures for real fuzzy numbers , 2002 .