Perfect GMV-Algebras

Perfect residuated lattice ordered monoids

“…Note that GM V -algebras, which are called "perfect GM V -algebras" in [6], are in fact strong perfect pseudo M V -algebras. We adopt the terminology of [22], and therefore by a strong perfect GM V -algebra we mean a symmetric perfect pseudo M V -algebra.…”

“…Note that in [22] and in [6], for GM V -algebras, the dual notions to those as "the order of an element", "local R -monoid", etc., were used. Nevertheless, we will use the fact that the operation ⊕ and in GM V -algebras are mutually dual and that the dual algebra to any GM V -algebra is a GM V -algebra as well, and hence we will apply the terminology of R -monoids also for GM V -algebras.…”

“…Recall that a GM V -algebra M is symmetric ( [5,6]) if x − = x ∼ for every x ∈ M . (Symmetric GM V -algebras were also studied in [26,27] under the name cyclic GM V -algebras.)…”

“…Recall that in contrast to bounded commutative R -monoids which have at least one state ( [12]), there are noncommutative R -monoids that do not admit states ( [9]). But it was shown that every linear pseudo BL-algebra ( [11]), as well as every perfect GM V -algebra ( [6,22]), admits a state. (Note that the existence of states on perfect GM V -algebras was explicitly proved in [6] only for symmetric perfect GM V -algebras, but, by the results in [22], it is true also for every perfect GM V -algebras.)…”

“…But it was shown that every linear pseudo BL-algebra ( [11]), as well as every perfect GM V -algebra ( [6,22]), admits a state. (Note that the existence of states on perfect GM V -algebras was explicitly proved in [6] only for symmetric perfect GM V -algebras, but, by the results in [22], it is true also for every perfect GM V -algebras.) Using the latter result, the authors [29] described a class of perfect bounded R -monoids having states.…”