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The paper presents some results on the logic psBL (pseudo-basic fuzzy logic, the generalization of BL not assuming commutativity of conjunction) and on the analogous logic psMTL -a non-commutative version of the monoidal t-norm logic MTL of Esteva and Godo. IntroductionThis paper is a free continuation of my paper [15], where the non-commutative version psBL of the basic fuzzy logic BL was presented. BL is the logic of continuous t-norms (see [17]) and has been generalized in various directions in recent past: I recall MTL -the logic of left-continuous t-norms [10,19] and the hoop logic BLH -the falsity free part of BL [11].Each of these logics has its algebraic counterpart -a variety of algebras of truth functions (BL-algebras for BL, MTL-algebras for MTL, basic hoops for BLH and pseudo-BL-algebras for psBL). Whereas in the case of BL and MTL the algebraic investigations followed the logical ones, in the case of the basic hoop logic and pseudo BL the algebraic development was first. In particular, psBL-algebras were introduced in [6, 7] and further developed and generalized in [12] without any logical consideration. The corresponding logic psBL was then introduced and analyzed in my [15]. The present paper relies substantially on the recent paper [20] by Kühr (showing that representable psBL-algebras form a variety).Non-commutative logics are relevant for computer science; see e.g. [1][2][3]. Non-commutative conjunctions are considered in fuzzy logic programming, see [22,23].Section 2 is rather short and formulates and proves consequences of Kühr's characterization of (subdirectly) representable psBL-algebras for logic. A stronger logic psBL r (r for representable) is introduced and shown to be strongly complete with respect to interpretations in linearly ordered psBL-algebras (psBL-chains). A natural example of a non-commutative ps-chain is presented in the appendix.Section 3 defines a non-commutative version psMTL of the logic MTL and its algebraic semantics -psMTLalgebras. Note that also here several algebraic results have preceded and are contained in [12], where the name ''weak psBL algebras'' is used (as well as ''weak BL-algebras'' for MTL-algebras). Our main aim here is to generalize Kühr's results to get a logic psMTL r which is shown to be strongly complete w.r.t. interpretations over psMTL-chains.
The paper presents some results on the logic psBL (pseudo-basic fuzzy logic, the generalization of BL not assuming commutativity of conjunction) and on the analogous logic psMTL -a non-commutative version of the monoidal t-norm logic MTL of Esteva and Godo. IntroductionThis paper is a free continuation of my paper [15], where the non-commutative version psBL of the basic fuzzy logic BL was presented. BL is the logic of continuous t-norms (see [17]) and has been generalized in various directions in recent past: I recall MTL -the logic of left-continuous t-norms [10,19] and the hoop logic BLH -the falsity free part of BL [11].Each of these logics has its algebraic counterpart -a variety of algebras of truth functions (BL-algebras for BL, MTL-algebras for MTL, basic hoops for BLH and pseudo-BL-algebras for psBL). Whereas in the case of BL and MTL the algebraic investigations followed the logical ones, in the case of the basic hoop logic and pseudo BL the algebraic development was first. In particular, psBL-algebras were introduced in [6, 7] and further developed and generalized in [12] without any logical consideration. The corresponding logic psBL was then introduced and analyzed in my [15]. The present paper relies substantially on the recent paper [20] by Kühr (showing that representable psBL-algebras form a variety).Non-commutative logics are relevant for computer science; see e.g. [1][2][3]. Non-commutative conjunctions are considered in fuzzy logic programming, see [22,23].Section 2 is rather short and formulates and proves consequences of Kühr's characterization of (subdirectly) representable psBL-algebras for logic. A stronger logic psBL r (r for representable) is introduced and shown to be strongly complete with respect to interpretations in linearly ordered psBL-algebras (psBL-chains). A natural example of a non-commutative ps-chain is presented in the appendix.Section 3 defines a non-commutative version psMTL of the logic MTL and its algebraic semantics -psMTLalgebras. Note that also here several algebraic results have preceded and are contained in [12], where the name ''weak psBL algebras'' is used (as well as ''weak BL-algebras'' for MTL-algebras). Our main aim here is to generalize Kühr's results to get a logic psMTL r which is shown to be strongly complete w.r.t. interpretations over psMTL-chains.
We study algebraic conditions when a pseudo MV-algebra is an interval in the lexicographic product of an Abelian unital ℓ-group and an ℓgroup that is not necessary Abelian. We introduce (H, u)-perfect pseudo MValgebras and strong (H, u)-perfect pseudo MV-algebras, the latter ones will have a representation by a lexicographic product. Fixing a unital ℓ-group (H, u), the category of strong (H, u)-perfect pseudo MV-algebras is categorically equivalent to the category of ℓ-groups. MV-algebra, strong (H, u)-perfect pseudo MV-algebra
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