On the relation between fuzzy max-Archimedean t-norm relational equations and the covering problem

On fuzzy relational equations and the covering problem

“…Please refer to Theorem 2 of [35] for the arithmetic mean; page 187 of [19] or Corollary 2 of [22] for the continuous Archimedean t-norm, and Theorem 2 of [39] for the ''fuzzy and'' with c<1. . Please refer to Markovskii [26], Li and Fang [19] and Lin [22] for algorithms for solving the covering problem. …”

“…Since then, FREs based on various compositions have been investigated. Some commonly compositions include max-min [4,5,12,18,30,32,40], max-product [3,23,25,26,31,36], max-Archimedean t-norm [22,34,37], max-t-norm [29,33] and maxarithmetic mean [16,35] compositions. Di Nola et al [8] indicated that if the solvability of max-continuous t-norm FREs is assumed, then the solution set for the FREs can be fully determined from a unique greatest solution and all minimal solutions, and the number of minimal solutions is always finite (see also [6,12]).…”

Fuzzy relational equations and the covering problem

“…Further, Markovskii proved that minimal solutions of system of equations with max-product composition correspond to irredundant coverings. Lin [17] extended Markovskii's work to fuzzy relational equations with max-Archimedean-t-norm composition. Further, Lin [18] investigated fuzzy relational equations with u-norm and transformed the problem of solving a system of fuzzy relational equations into covering problem.…”

“…In fact, fuzzy relational equations can be categorized based on many different compositions. Some common compositions include max-min [4,7,12,25], max-product [7,23], max-Archimedean tnorm [2,4,7,11,17,19], inf-α [7,15,16] and inf-α T [2,7,30] compositions. Wu [29] and Di Nola et al [6] found that in fuzzy relational calculus and reasoning inf-α composition is better.…”

“…In 2012, Shieh [26] developed an efficient algorithm for finding minimal coverings. In fact, Lin [17,18] and Shieh [26] discussed the relations between the minimal solutions of the equations and the irredundant coverings. However, fuzzy relational equations with inf-implication composition have no minimal solutions, there are maximal solutions of the equations, in this paper, we discuss fuzzy relational equations with inf-implication composition on [0,1].…”