Pseudo EQ-algebras were introduced by X.L. Xin as possible algebraic semantics for fuzzy type theory. The main goal of this paper is to investigate pseudo EQ-algebras with internal states. To begin with, we introduce the notion of pseudo EQ-algebras with internal state(simplify, ISPEQ-algebra) and study its properties. Moreover, we discuss the relation between ISPEQ-algebras and states on pseudo EQ-algebras. In the following, we study state filters(simplify S-fliters) and state prefilters(simplify S-prefliters) of ISPEQ-algebras. We construct ISPEQ-algebras from S-fliters and derive congruence relation θ of ISPEQ-algebras. In the end, we give the generating formula of prefilters on residuated pseudo EQ-algebras and based on that we get the generating formula of S-prefilters on residuated pseudo EQ-algebras.
FlaminioT. and MontagnaF., An algebraic approach to states on MV-algebras, in:V. Novak(ED.), Fuzzy logic 2, Proc. of the 5th USFLAT Conf., Sept., 2(11-14) (2007), 201–206
12.
FlaminioT., MontagnaF., MV-algebras with internal states andprobabilistic fuzzy logic, Int J of Approx Reason50(2009), 138–152.
13.
GeorgescuG. and IorgulescuA., Pseudo-MV algebras: a non-commutative extension of MV-algebras. In: Proc. 4th International Symposium of Economic Informatics, INFOREC Printing House, Bucharest, 1999, 961–968.
14.
GeorgescuG. and IorgulescuA., Pseudo-BL algebras: a non-commutative extension of BL-algebras. In: Abstracts of the 5th International Conference FSTA 2000, Slovakia, pp. 90–92.
15.
GeorgescuG. and IorgulescuA., Pseudo-MV algebras, Mult ValuedLog6 (2001), 95–135.
16.
GeorgescuG. and IorgulescuA., Pseudo-BCK algebras: an extension of BCK-algebras, In: Proc.DMTCS01: Combinatorics, Computability and Logic, 97–114, Springer, London, 2001.
HeP.F., XinX.L., YangY.W., On state residuated lattices, Soft Comput19 (2015), 2083–2094.
19.
HeP.F., ZhaoB., XinX.L., States and internal states on semihoops, Soft Comput21 (2017), 2941–2957.
20.
JeneiS., KąäorąäodiL., Pseudo equalityalgebras, Arch Math Log52 (2013), 469–481.
21.
KadjiA., LeleC., TongaM., Some classes of pseudo-residuatedlattices, Afr Mat27 (2016), 1147–1178.
22.
LeusteanI., Non-commutative Lukasiewicz propositional logic, Arch Math Log45 (2006), 191–213.
23.
LiuL.Z., On the existence of states on MTL-algebras, Inf Sci220 (2013), 559–567.
24.
MundiciD., Averaging the truth value in Lukasiewicz logic, Studia Logic55 (1995), 113–127.
25.
NovákV., On fuzzy type theory, Fuzzy Sets and Systems149(2) (2005), 235–273.
26.
NovákV., Fuzzy type theory as higher-order fuzzy logic, In: Proceedings of the 6th international conference on intelligent technologies, Bangkok, Thailand, (2005).
27.
NovákV., EQ-algebra, Primary concepts and properties, Int Joint, Czech Republic-Japan and Taiwan-Japan Symposium, Kitakyushu, Japan, 2006, 219–223.
28.
RiečanB., On the probability on BL-algebras, Acta MathNitra4 (2000), 3–13.
