Boolean-like algebras. (English) Zbl 1284.06033

The aim of the paper is to look for an answer to the following questions formulated in the introduction:
– What is so special about Boolean algebra that is responsible for its nice behavior?
– Given a similarity type, can we always find a class of algebras of this type that displays Boolean-like features?
– What does it mean, for an algebra of a given type that may not exhibit such desirable properties, to have at least a subset of Boolean elements that behave well?
To answer these questions, the authors introduce the notion of a Boolean-like algebra, and then of an even more general notion of a semi-Boolean-like algebra, as a generalization of a Boolean algebra to an arbitrary type. They use Vaggione’s concept of a central element in a double-pointed algebra and the concept of a Church algebra. A central element is an element that induces a pair of complementary factor congruences, and a Church algebra is a double-pointed algebra with constants \(0\) and \(1\) and a ternary operation \(q(x,y,z)\) such that \(q(1,x,y) = x\) and \(q(0,x,,y) = y\). A semi-Boolean algebra is a Church algebra (of an arbitrary type) with the operation \(q\) satisfying certain additional equations. A Boolean-like algebra is defined as a semi-Boolean-algebra satisfying one more additional equation. In a Boolean-like algebra, all elements are central. The authors investigate properties of (semi-)Boolean-like algebras and provide a number of characterizations of some classes of such algebras. The main results include the following:
– A variety of Church algebras is a semi-Boolean-like variety if an only if it is \(c\)-permutable and \(B\)-semisimple.
– A variety of double-pointed algebras is a Boolean-like variety if and only if it is a discriminator variety with all subdirectly irreducible members having precisely two element.
– A variety of double-pointed algebras is a discriminator variety if and only if it is 0-regular and idempotent semi-Boolean-like.
The results tie in nicely with three research streams: (weak) Boolean product representations, discriminator varieties and noncomutative lattice theory, and an algebraic investigation of the if-then-else construct.


06E75 Generalizations of Boolean algebras
08B05 Equational logic, Mal’tsev conditions
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[1] Aglianò P., Ursini A.: On subtractive varieties II: General properties. Algebra Universalis 36, 222-259 (1996) · Zbl 0902.08010
[2] Aglianò P., Ursini A.: On subtractive varieties III: From ideals to congruences. Algebra Universalis 37, 296-333 (1997) · Zbl 0906.08005
[3] Aglianò P., Ursini A.: On subtractive varieties IV: Definability of principal ideals. Algebra Universalis 38, 355-389 (1997) · Zbl 0934.08002
[4] Barbour G.D., Raftery J.G.: On the degrees of permutability of subregular varieties. Czechoslovak Math. J. 47, 317-325 (1997) · Zbl 0927.08001
[5] Barendregt H.P.: The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984) · Zbl 0551.03007
[6] Bergman G.M.: Actions of Boolean rings on sets. Algebra Universalis 28, 153-187 (1991) · Zbl 0692.18003
[7] Bigelow D., Burris S.: Boolean algebras of factor congruences. Acta Sci. Math. (Szeged) 54, 11-20 (1990) · Zbl 0714.08001
[8] Bignall R.J., Leech J.: Skew Boolean algebras and discriminator varieties. Algebra Universalis 33, 387-398 (1995) · Zbl 0821.06013
[9] Blok, W.J., Pigozzi, D.: Algebraizable logics. Mem. Amer. Math. Soc. 77, no. 396 (1989) · Zbl 0664.03042
[10] 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
[11] Blok W.J., Pigozzi D.: On the structure of varieties with equationally definable principal congruences IV. Algebra Universalis 31, 1-35 (1994) · Zbl 0817.08005
[12] Blok W.J., Raftery J.G.: Assertionally equivalent quasivarieties. Internat. J. Algebra Comput. 18, 589-681 (2008) · Zbl 1148.08002
[13] Burris S.N., Sankappanavar H.P.: A Course in Universal Algebra. Springer, Berlin (1981) · Zbl 0478.08001
[14] Burris S.N., Werner H.: Sheaf constructions and their elementary properties. Trans. Amer. Math. Soc. 248, 269-309 (1979) · Zbl 0411.03022
[15] Burris S.N., Werner H.: Remarks on Boolean products. Algebra Universalis 10, 333-344 (1980) · Zbl 0454.08007
[16] Comer S.: Representations by algebras of sections over Boolean spaces. Pacific J. Math. 38, 29-38 (1971) · Zbl 0219.08002
[17] Czelakowski J.: Equivalential logics I. Studia Logica 45, 227-236 (1981) · Zbl 0476.03032
[18] Dicker R.M.: A set of independent axioms for Boolean algebra. Proc. London Math. Soc. 3, 20-30 (1963) · Zbl 0106.24301
[19] Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam (2007) · Zbl 1171.03001
[20] Givant S., Halmos P.: Introduction to Boolean Algebras. Springer, Berlin (2009) · Zbl 1168.06001
[21] Hagermann J., Hermann C.: A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity. Arch. Math. 32, 234-245 (1979) · Zbl 0419.08001
[22] Jackson M., Stokes T.: Semigroups with if-then-else and halting programs. Internat. J. Algebra Comput. 19, 937-961 (2009) · Zbl 1203.08005
[23] Jipsen P.: Generalizations of Boolean products for lattice-ordered algebras. Ann. Pure Appl. Logic 161, 228-234 (2009) · Zbl 1181.06005
[24] Jónsson B.: Congruence distributive varieties. Math. Japon. 42, 353-402 (1995) · Zbl 0841.08004
[25] Kowalski T., Paoli F.: Joins and subdirect products of varieties. Algebra Universalis 65, 371-391 (2011) · Zbl 1233.08007
[26] Kowalski T., Paoli F., Spinks M.: Quasi-subtractive varieties. J. Symbolic Logic 76, 1261-1286 (2011) · Zbl 1254.03119
[27] Kozen D.C.: On Hoare Logic and Kleene algebra with Tests. ACM Trans. Comput. Log. 1, 60-76 (2000) · Zbl 1365.68326
[28] Ledda A., Konig M., Paoli F., Giuntini R.: MV algebras and quantum computation. Studia Logica 82, 245-270 (2006) · Zbl 1102.06010
[29] Ledda, A., Paoli, F., Spinks, M., Kowalski, T.: Quasi-subtractive varieties: Open filters, congruences, and the commutator (submitted) · Zbl 1335.08004
[30] Manzonetto, G., Salibra, A.: Boolean algebras for lambda calculus. In: Proceedings of the 21th IEEE Symposium on Logic in Computer Science (LICS 2006), pp. 317-326. IEEE Computer Society (2006)
[31] Manzonetto, G., Salibra, A.: From λ-calculus to universal algebra and back. In: Proceedings of MFCS’08 Symposium. Lecture Notes in Computer Science, vol. 5162, pp. 479-490. Springer (2008) · Zbl 1173.03302
[32] Manzonetto, G., Salibra, A.: Lattices of equational theories as Church algebras. In: Drossos, C., et al. (eds.) Proceedings of the Seventh Panhellenic Logic Symposium, pp. 117-121. Patras University Press (2009) · Zbl 0714.08001
[33] Manzonetto G., Salibra A.: Applying universal algebra to lambda calculus. J. Logic Comput. 20, 877-915 (2010) · Zbl 1216.03030
[34] McCarthy, J.: A basis for a mathematical theory of computation. In: Braffort, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, pp. 33-70. North-Holland, Amsterdam (1963) · Zbl 0902.08010
[35] McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. I. Wadsworth Brooks, Monterey (1987) · Zbl 0611.08001
[36] Peirce, R.S.: Modules over Commutative Regular Rings. Modules over commutative regular rings. Memoirs of the American Mathematical Society, no. 70. American Mathematical Society, Providence (1967) · Zbl 0934.08002
[37] Pratt, V.R.: Dynamic Algebras: Examples, Constructions, Applications. Technical Report MIT/LCS/TM-138. M.I.T. Laboratory for Computer Science (1979) · Zbl 0752.03033
[38] Ursini A.: On subtractive varieties I. Algebra Universalis 31, 204-222 (1994) · Zbl 0799.08010
[39] Vaggione D.: Varieties in which the Pierce stalks are directly indecomposable. J. Algebra 184, 424-434 (1996) · Zbl 0868.08003
[40] Vaggione D.: \[{\mathcal{V}}\] with factorable congruences and \[{\mathcal{V} = I \Gamma^a(\mathcal{V}_{DI})}\] imply \[{\mathcal{V}}\] is a discriminator variety. Acta Sci. Math. (Szeged) 62, 359-368 (1996) · Zbl 0880.08007
[41] Werner, H.: Discriminator Algebras. Studien zur Algebra und ihre Anwendungen, vol. 6. Akademie, Berlin (1978) · Zbl 0374.08002
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