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Varieties of BL-algebras. I: General properties. (English) Zbl 1034.06009
The aim of the paper is to show some facts and techniques that are useful in order to describe the lattice of subvarieties of BL-algebras. An algebra \((A,\to,\cdot,0,1)\) is called a BL-algebra if \((A,\cdot,1)\) is a commutative monoid and the following identities are satisfied: \(x\to x= 1\), \(x\cdot(x\to y)= y\cdot(y\to x)\), \(x\to (y\to z)= (x\cdot y)\to z\), \(0\to x=1\) and \(((x\to y)\to z)\to (((y\to x)\to z)\to z)= 1\). Some special cases of BL-algebras are introduced; for example a BL-algebra satisfying the equation \(x\cdot x= x\) is called a Gödel algebra. The results of the paper include a description of subalgebras and homomorphic images of totally ordered BL-algebras and a characterization of totally ordered BL-algebras that generate the variety of all BL-algebras and other results.

MSC:
06D35 MV-algebras
08B15 Lattices of varieties
08A30 Subalgebras, congruence relations
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[1] P. Aglianó, I.M.A. Ferreirim, F. Montagna, Basic hoops: an algebriac study of continuous t-norms, 1999, preprint.
[2] P. Aglianó, G. Panti, Geometrical methods in Wajsberg hoops, 2000, preprint.
[3] Blok, W.J; Ferreirim, I.M.A, Hoops and their implicational reducts, (1993), Algebraic Logic, Banach Center pub. 28 Warsaw · Zbl 0848.06013
[4] Blok, W.J; Ferreirim, I.M.A, On the structure of hoops, Algebra universalis, 43, 233-257, (2000) · Zbl 1012.06016
[5] Blok, W.J; Pigozzi, D, On the structure of varieties with equationally definable principal congruences III, Algebra universalis, 32, 545-608, (1994) · Zbl 0817.08004
[6] Blok, W.J; Raftery, J, Varieties of commutative residuated integral pomonoids and their residuation subreducts, J. algebra, 190, 280-328, (1997) · Zbl 0872.06007
[7] Bosbach, B, Komplementäre halbgruppen. axiomatik und arithmetik, Fund. math., 64, 257-287, (1969) · Zbl 0183.30603
[8] J.R. Büchi, T.M Owens, Complemented monoids and hoops, 1975, unpublished manuscript.
[9] S. Burris, H.P. Sankappanavar, A course in universal algebra, Graduate texts in Mathematics, Springer, Berlin, 1981. · Zbl 0478.08001
[10] Chang, C.C, Algebraic analysis of many-valued logic, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704
[11] Cignoli, R; Esteva, F; Godo, L; Torrens, A, Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft comput., 4, 106-112, (2000)
[12] Cignoli, R; Torrens, A, An algebraic analysis of product logic, Mult. val. logic, 5, 45-65, (2000) · Zbl 0962.03059
[13] Di Nola, A; Lettieri, A, Equational characterization of all varieties of MV algebras, J. algebra, 22, 463-474, (1999) · Zbl 0946.06009
[14] I.M.A. Ferreirim, On varieties and quasi varieties of hoops and their reducts, Ph.D. Thesis, University of Illinois at Chicago, 1992.
[15] Gispert, J, Universal classes of MV-chains with applications to many-valued logics, Math. log. quart., 48, 581-601, (2002) · Zbl 1049.06009
[16] Gispert, J; Mundici, D; Torrens, A, Ultraproducts of \( Z\) with an application to many-valued logics, J. algebra, 219, 214-233, (1999) · Zbl 0937.06008
[17] Gumm, P; Ursini, A, Ideals in universal algebra, Algebra universalis, 19, 45-54, (1984) · Zbl 0547.08001
[18] P. Hájek, Metamathematics of Fuzzy Logic, Trends in Logic-Studia Logica Library no. 4 Kluwer Academic Publishers, Dordercht/Boston/London, 1998.
[19] Hájek, P, Basic fuzzy logic and BL algebras, Soft comput., 2, 3, 124-128, (1998)
[20] Hecht, T; Katrinak, T, Equational classes of relative stone algebras, Notre dame J. formal logic, 13, 248-254, (1972) · Zbl 0212.01601
[21] Komori, Y, Super-łukasiewicz implicational logics, Nagoya math. J., 84, 1119-1133, (1981)
[22] Mostert, P.S; Shields, A.L, On the structure of semigroups on a compact manifold with boundary, Ann. math., 65, 117-143, (1957) · Zbl 0096.01203
[23] Mundici, D, Interpretation of AFC*-algebras in łukasiewicz sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059
[24] Panti, G, Varieties of MV algebras, J. appl. non-classical logic, 9, 141-157, (1999) · Zbl 1031.06010
[25] Torrens, A, Cyclic elements in MV-algebras and in post algebras, Math. log. quart., 40, 431-444, (1994) · Zbl 0815.06009
[26] Ursini, A, On subtractive varieties I, Algebra universalis, 31, 204-222, (1994) · Zbl 0799.08010
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