A Semiring-based study of judgment matrices: properties and models

A further discussion of “A Semiring-based study of judgment matrices: properties and models” [Information Sciences 181 (2011) 2166–2176]

“…In section 3.5 of [11], in order to present results from the perspective of semiring isomorphisms, the author considers multiplicative PCMs with entries p ij 2 1 a ; a  à with a > 1 and additive PCMs with entries a ij 2 ½0; 1, that are subsets of underlying sets of K  ¼ ð½0; þ1Þ; È Â ;  ; e  ; e Â Þ and K þ ¼ ð½À1; þ1Þ; È þ ; þ ; e þ ; e þ Þ, respectively. By Remark 2.2, multiplicative alo-group and fuzzy alo-group are isomorphic between them; in particular the function [3]:…”

“…ðK; ÈÞ is a commutative monoid with identity element e: Main results of [11] can be summarized as follows:…”

“…As a consequence, multiplicative and additive reciprocity in (2.1) and (2.2) can be generalized to a PCM defined over a general semiring ðK; È; ; e; eÞ, in the following way: However, the author in [11] does not aim at generalizing the approach to a general semiring, he introduces the semirings K  and K þ for dealing with multiplicative and additive PCMs respectively; moreover, both each p 2 ð0; þ1Þ (or in ½ 1 a ; a with a > 1) and each a 2 ½0; 1 has the inverse element with respect to  and þ respectively, thus they can be entries of multiplicative and additive PCMs.…”

“…A comparison between K  ¼ ð½0; þ1Þ; È Â ;  ; e  ; e Â Þ and 0; þ1½= ð0; þ1½; Á; 6Þ for dealing with PCMs Firstly, the case described in section 3.2 of [11], i.e. p ij 2 ð0; þ1Þ, is analyzed; thus, by considering K  and 0 þ 1½ for the discussion of PCMs, the following considerations can be done:…”

“…In Ref. [11], the author proposes two special semirings to discuss PCMs; the approach is summarized in Section 2.1; in Section 2.2, abelian linearly ordered groups are shown for dealing with PCMs (see [3][4][5][6][7][8] for further details). In Section 3 a comparison between the two approaches is performed in order to choose the more convenient one.…”