Abstract
A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a “classical” ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.
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References
Agliano, P. andUrsini, A.,On some Ideal Basis Theorems, AILA PREPRINTS n.10, 1991.
Agliano, P. andUrsini, A.,Ideals and other generalizations of congruence classes, J. of Austr. Math. Soc. (Series A)53 (1992), 103–115.
Beutler, E.,Kaiser, H.,Matthiessen, G. andTimm, J.,Biduale algebren, Mathematik Arbeitspapiere Nr. 21, Universität Bremen, 1979.
Büchi, J. R. andOwens, T. M.,Skolem rings and their varieties, The collected works of J. Richard Büchi, Saunders Maclane and Dirk Siefkes eds, Springer Verlag, New York, N.Y., 1990.
Duda, J.,Arithmeticity at 0, Czech. Math. J.37 (112) (1987), 197–206.
Freese, R. andMcKenzie, R.,Commutator Theory for Congruence Modular Varieties, London Math. Soc. Lecture Note Series, Cambridge University Press, Cambridge, 1987.
Gumm, H. P. andUrsini, A.,Ideals in Universal Algebra, Algebra Universalis19 (1984), 45–54.
McKenzie, R., McNulty, G. andTaylor, W.,Algebras, Lattices, Varieties, Volume I, Wadsworth and Brooks Cole, Monterey, California, 1987.
Troelstra, A. S.,Lectures on Linear logic, Institute for Language, Logic and Information, Amsterdam, December 1990 (Errata and supplementibidem, March 1991).
Ursini, A.,Sulle varietá di algebre con una buona teoria degli ideali, Boll. U.M.I.6 (1972), 90–95.
Ursini, A.,Osservazioni sulla varietá BIT, Boll. U.M.I.8 (1973), 205–211.
Ursini, A.,Ideals and their calculus I, Rapporto Matematico n. 41, Università di Siena (1981).
Ursini, A.,Prime Ideals in Universal Algebra, Acta Univ. Carol. Math, et Phys.25 (1984), 75–87.