Abstract
In this paper we investigate the properties of the relative negations in non-commutative residuated lattices and their applications. We define the notion of a relative involutive FL-algebra and we generalize to relative negations some results proved for involutive pseudo-BCK algebras. The relative locally finite IFL-algebra is defined and it is proved that an interval algebra of a relative locally finite divisible IFL-algebra is relative involutive. Starting from the observation that in the definition of states, the standard MV-algebra structure of [0, 1] intervenes, there were introduced the states on bounded pseudo-BCK algebras, pseudo-hoops and residuated lattices with values in the same kind of structures and they were studied under the name of generalized states. For the case of commutative residuated lattices the generalized states were studied in the sense of relative negation. We define and study the relative generalized states on non-commutative residuated lattices. One of the main results consists of proving that every order-preserving generalized Bosbach state is a relative generalized Riečan state. Some conditions are given for a relative generalized Riečan state to be a generalized Bosbach state. Finally, we develop a concept of states on IFL-algebras.
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References
Buşneag C (2010) States on Hilbert algebras. Stud Logica 94:177–188
Cignoli R, Torrens A (2004) Glivenko like theorems in natural expansions of BCK-logic. Math Logic Q 50:111–125
Cignoli R, Torrens A (2006) Free algebras in varieties of Glivenko MTL-algebras satisfying the equation 2(x 2) = (2x)2. Studia Logica 83:157–181
Ceterchi R (2001) Pseudo-Wajsberg algebras. Mult Valued Logic 6:67–88
Ciungu LC (2006) Classes of residuated lattices. An Univ Craiova Ser Mat Inf 33:189–207
Ciungu LC (2008) Bosbach and Riečan states on residuated lattices. J Appl Funct Anal 2:175–188
Ciungu LC (2009) The radical of a perfect residuated structure. Inform Sci 179:2695–2709
Ciungu LC (2010) On pseudo-BCK algebras with pseudo-double negation. An Univ Craiova Ser Mat Inf 37(1):19–26
Ciungu LC, Georgescu G, Mureşan C (2013) Generalized Bosbach states: part I. Arch Math Logic. doi:10.1007/s00153-012-0319-2
Ciungu LC, Kühr J (2013) New probabilistic model for pseudo-BCK algebras and pseudo-hoops. J Mult Valued Logic Soft Comput 20:373–400
Di Nola A, Georgescu G, Iorgulescu A (2002a) Pseudo BL-algebras: part I. Mult Valued Logic 8:673–714
Di Nola A, Georgescu G, Iorgulescu A (2002b) Pseudo BL-algebras: part II. Mult Valued Logic 8:715–750
Dvurečenskij A, Hyčko M (2006) Algebras on subintervals of BL-algebras, pseudo-BL algebras and bounded residuated Rℓ-monoids. Math Slovaca 56:125–144
Flondor P, Georgescu G, Iorgulescu A (2001) Pseudo-t-norms and pseudo-BL algebras. Soft Comput 5:355–371
Gaifman H (1964) Concerning measures on first-order calculi. Israel J Math 2:1–18
Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logic. Elsevier, The Netherlands
Georgescu G, Leuştean L (2002) Some classes of pseudo-BL algebras. J Aust Math Soc 73:127–153
Hájek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publications, Dordrecht
Iorgulescu A (2008) Algebras of logic as BCK algebras,. Editura ASE, Bucharest
Jipsen P, Tsinakis C (2002) A survey of residuated lattices. In: Martinez J (ed) Ordered algebraic structures. Kluwer Academic Publishers, Dordrecht, pp 19–56
Scott D, Krauss P (1966) Assigning probabilities to logical formulas. In: Hintikka J, Suppes P (eds) Aspects of inductive logic. North-Holland, Amsterdam, pp 219–264
Turunen E, Mertanen J (2008) States on semi-divisible generalized residuated lattices reduce to states on MV-algebras. Fuzzy Sets Syst 22:3051–3064
Zhou H, Zhao B (2012) Generalized Bosbach and Riečan states based on relative negations in residuated lattices. Fuzzy Sets Syst 187:33–57
Wille AM (2006) Residuated structures with involution. PhD thesis, Darmstadt Technical University
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The author is very grateful to the referees for the valuable suggestions in obtaining the final form of this paper.
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Communicated by A. Dvurečenskij.
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Ciungu, L.C. Relative negations in non-commutative fuzzy structures. Soft Comput 18, 15–33 (2014). https://doi.org/10.1007/s00500-013-1054-2
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DOI: https://doi.org/10.1007/s00500-013-1054-2