Abstract
This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: {product, inverse}, {division}, {double division, identity}, {double division, inverse}, {division, identity}, and {division, inverse}. In all but two of the twelve corresponding theories, we present either the first single axioms known to us or single axioms shorter than those previously known to us. The automated theorem-proving program OTTER was used extensively to construct sets of candidate axioms and to search for and find proofs that given candidate axioms are in fact single axioms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Higman, G. and Neumann, B. H., ‘Groups as groupoids with one law’,Publicationes Mathematicae Debrecen 2, 215–227 (1952).
Hsiang, J. and Rusinowitch, M., ‘On word problems in equational theories’, in T. Ottmann (ed),Proceedings of 14th ICALP, Lecture Notes in Computer Science, Vol. 267, Springer-Verlag, New York, 1987, pp. 54–71.
Kapur, D. and Zhang, H., ‘Proving equivalence of different axiomatizations of free groups’,Journal of Automated Reasoning 4(3), 331–352 (1988).
Knuth, D. and Bendix, P., ‘Simple word problems in universal algebras’, in J. Leech (ed),Computational Problems in Abstract Algebras, Pergamon Press, Oxford, 1970, pp. 263–297.
Kunen, K., Sept.–Oct. 1991, Correspondence by electronic mail.
Kunen, K., ‘Single axioms for groups’,J. Automated Reasoning 9(3), 291–308 (1992).
Lankford, D., ‘Canonical inference’, Technical Report ATP-32, Dept. of Computer Sciences, University of Texas, Austin, TX, 1975.
McCune, W., ‘OTTER 2.0 Users Guide’, Technical Report ANL-90/9, Argonne National Laboratory, Argonne, IL, March 1990.
McCune, W., ‘Proofs for groups and Abelian group single axioms’, Technical Memo ANL/MCS-TM-156, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, October 1991.
McCune, W., ‘Single axioms for the left group and right group calculi’, Preprint MCS-P219-0391, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 1991. To appear inNotre Dame J. Formal Logic.
McCune, W., ‘What's new in OTTER 2.2’, Technical Memo ANL/MCS-TM-153, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, July 1991.
McCune, W., ‘Automated discovery of new axiomatizations of the left group and right group calculi’,Journal of Automated Reasoning 9(1), 1–24 (1992).
Neumann, B. H., ‘Another single law for groups’,Bull. Australian Math. Soc. 23, 81–102 (1981).
Neumann, B. H., ‘Yet another single law for groups’,Illinois Journal of Mathematics 30(2), 295–300 (1986).
Slaney, J., ‘FINDER, finite domain enumerator: Version 1.0 notes and guide’, Technical Report TR-ARP-10/91, Automated Reasoning Project, Australian National University, Canberra, Australia, 1991.
Tarski, A., ‘Ein Beitrag zur Axiomatik der Abelschen Gruppen’,Fundamenta Mathematicae 30, 253–256 (1938).
Tarski, A., ‘Equational logic and equational theories of algebras’, in K. Schütte (ed),Contributions to Mathematical Logic, North-Holland, Amsterdam, 1968, pp. 275–288.
Author information
Authors and Affiliations
Additional information
This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
Rights and permissions
About this article
Cite this article
McCune, W.W. Single axioms for groups and Abelian groups with various operations. J Autom Reasoning 10, 1–13 (1993). https://doi.org/10.1007/BF00881862
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00881862