Free Term Algebras

We interoduce a new characterization of algebras of normal forms of term rewriting systems [35] as algerbras of term free in itself (any function from free generators into the algebra generates endomorphism of the algebra). Introduced algebras are free in classes of algebras satisfying some sets of equalities. Their universes are subsets of all terms and the denotations of operation symbols are partially identical with the operations of construction of terms. These algebras are compiler algebras requiring some equalities of terms, e.g., associativity of addition.This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Faculty of Computer Science, Białystok Technical University, Wiejska 45A, 15-351 Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991.Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397-402, 1991.Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195-204, 1992.Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.Grzegorz Bancerek. Reduction relations. Formalized Mathematics, 5(4):469-478, 1996.Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185-190, 1996.Grzegorz Bancerek. Terms over many sorted universal algebra. Formalized Mathematics, 5(2):191-198, 1996.Grzegorz Bancerek. Translations, endomorphisms, and stable equational theories. Formalized Mathematics, 5(4):553-564, 1996.Grzegorz Bancerek. Algebra of morphisms. Formalized Mathematics, 6(2):303-310, 1997.Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279-287, 1997.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.[16] Grzegorz Bancerek and Artur Korniłowicz. Yet another construction of free algebra. Formalized Mathematics, 9(4):779-785, 2001.Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91-101, 1993.Grzegorz Bancerek and Piotr Rudnicki. The set of primitive recursive functions. Formalized Mathematics, 9(4):705-720, 2001.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Ewa Burakowska. Subalgebras of many sorted algebra. Lattice of subalgebras. Formalized Mathematics, 5(1):47-54, 1996.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Artur Korniłowicz. Some basic properties of many sorted sets. Formalized Mathematics, 5(3):395-399, 1996.Artur Korniłowicz. Equations in many sorted algebras. Formalized Mathematics, 6(3):363-369, 1997.Małgorzata Korolkiewicz. Homomorphisms of many sorted algebras. Formalized Mathematics, 5(1):61-65, 1996.Małgorzata Korolkiewicz. Many sorted quotient algebra. Formalized Mathematics, 5(1):79-84, 1996.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Adam Naumowicz. On Segre’s product of partial line spaces. Formalized Mathematics, 9(2):383-390, 2001.Andrzej Nedzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67-74, 1996.D.M. Gabbay S. Abramsky and T.S.E. Maibaum, editors. Handbook of Logic inComputer Science, chapter Term Rewriting Systems, pages 1-116. Oxford University Press, New York, 1992. http://www.informatik.uni-bremen.de/agbkb/lehre/-rbs/texte/Klop-TR.pdf.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.Andrzej Trybulec. A scheme for extensions of homomorphisms of many sorted algebras. Formalized Mathematics, 5(2):205-209, 1996.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990

Automorphisms of Cuntz-Krieger algebras

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda} (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras.Comment: 26 page