zbMATH — the first resource for mathematics

Free algebras in varieties of Stonean residuated lattices. (English) Zbl 1142.06003
The variety of Stonean residuated lattices consists of all bounded (integral, commutative) residuated lattices $$\langle A, *, \rightarrow, \vee, \wedge, \top, \perp \rangle$$ that satisfy the identity $$\neg \, x \vee \neg \neg \,x = \top$$, where $$\neg \, x := x \rightarrow \, \perp$$. A key feature of a Stonean residuated lattice is that double negation is a retract onto its Boolean skeleton. This paper presents a weak Boolean product representation of the free algebras in varieties of Stonean residuated lattices. Given a variety $$V$$ of Stonean residuated lattices and a generating set $$X$$, let $$V^*$$ be the variety of all residuated lattices that become members of $$V$$ when a least element is appended. Then the $$V$$-free algebra over $$X$$ is represented as a weak Boolean product over the Cantor space $$2^{|X|}$$ of the family of $$V^*$$-free algebras over subsets of $$X$$, with a least element appended.

MSC:
 06F05 Ordered semigroups and monoids 08B20 Free algebras
Full Text:
References:
 [1] Bigelow D, Burris S (1990) Boolean algebras of factor congruences. Acta Sci Math 54:11–20 · Zbl 0714.08001 [2] Burris S, Werner H (1979) Sheaf constructions and their elementary properties. Trans Am Math Soc 248:269–309 · Zbl 0411.03022 [3] Busaniche M, Cignoli R (2006) Free algebras in varieties of BL- algebras generated by a BL n -chain. J Aust Math Soc 80:419–439 · Zbl 1094.03058 [4] Cignoli R, Torrens A (1996) Boolean products of MV-algebras: hypernormal MV-algebras. J Math Anal Appl 99:637–653 · Zbl 0849.06012 [5] Cignoli R, Torrens A (2000a) An algebraic analysis of product logic. Mult Valued Log 5:45–65 · Zbl 0962.03059 [6] Cignoli R, Torrens A (2000b) Free stone algebras. Discrete Math 222:251–257 · Zbl 0982.06010 [7] Cignoli R, Torrens A (2002) Free algebras in varieties of BL-algebras with a Boolean retract. Algebra Univers 48:55–79 · Zbl 1058.03077 [8] Cignoli R, Torrens A (2003) Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic. Arch Math Log 42:361–370 · Zbl 1025.03018 [9] Cignoli R, Torrens A (2004) Glivenko like theorems in natural expansions of BCK-logic. Math Log Q 50:111–125 · Zbl 1045.03026 [10] Cignoli R, Torrens A (2006) Free algebras in varieties of Glivenko MTL-algebras satisfying the equation 2(x 2) = (2x)2. Stud Log 83:157–181 · Zbl 1115.06006 [11] Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124:271–288 · Zbl 0994.03017 [12] Esteva F, Gispert J, Godo L, Montagna F (2002) On the standard and rational completeness of some axiomatic extensions of monoidal t-norm based logic. Stud Log 71:199–226 · Zbl 1011.03015 [13] Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht, Boston, London · Zbl 0937.03030 [14] Höhle U (1995) Commutative, residuated l-monoids. In: Höhle U, Klement EP (eds) Non-classical logics and their applications to fuzzy subsets: a handbook on the mathematical foundations of fuzzy set theory. Kluwer, Boston, pp 53–106 [15] Horn A (1969) Free L-algebras. J Symb Log 34:457–480 [16] Jónsson B (1995) Congruence distributive varieties. Math Jpn 42:353–401 · Zbl 0841.08004 [17] Kowalski T, Ono H (2001) Residuated lattices: an algebraic glimpse at logics without contraction. Preliminary report [18] Monteiro AA (1980) Sur les algèbres de Heyting symmétriques. Port Math 39:1–237 [19] Ono H (2003) Substructural logics and residuated lattices: an introduction. In: Hendricks VF, Malinowski J (eds) Trends in logic: 50 years of studia logica. Kluwer, Dordrecht, Boston, London, pp 177–212
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.