Abstract
All varieties of idempotent semigroups have been classified with respect to the unification types of their defining sets of identities. With the exception of eight finitary unifying theories, they are all of unification type zero. This yields countably many examples of theories of this type which are more “natural” than the first example constructed by Fages and Huet.
The lattice of all varieties of idempotent semigroups is a sublattice of the lattice of all varieties of orthodox bands of groups, and this lattice is a sublattice of the lattice of all varieties of completely regular semigroups. The proof which was used to establish the result for the varieties of idempotent semigroups of type zero can—with some modifications—also be applied to the larger lattice of all varieties of completely regular semigroups. This shows that type zero is not an exception, but rather common for varieties of semigroups.
To establish the results for the eight exceptional finitary varieties of idempotent semigroups we have developed a method which under certain conditions allows to deduce the unification type of a join of varieties from the types of the varieties participating in this join. This method can also be employed for varieties of orthodox bands of abelian groups. Any variety of orthodox bands of abelian groups is the join of a variety of idempotent semigroups and a variety of abelian groups. It turns out that the unification type of such a join is just the type of the variety of idempotent semigroups taking part in this join.
The emphasis of the paper is on describing the tools necessary for proving all the mentioned results.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
F. Baader, “The Theory of Idempotent Semigroups is of Unification Type Zero,” J. Automated Reasoning 2, 1986.
F. Baader, “Unification in Varieties of Idempotent Semigroups,” Semigroup Forum 36, 1987.
F. Baader, “A Note on Unification Type Zero,” Information Processing Letters 27, 1988.
F. Baader, “Characterizations of Unification Type Zero,” Proceedings of the Third International Conference on Rewriting Techniques and Applications, Springer LNCS 355, 1989.
F. Baader, Unifikation und Reduktionssysteme für Halbgruppenvarietäten, Dissertation, Institut für Mathematische Maschinen und Datenverarbeitung, Universität Erlangen-Nürnberg, 1989.
F. Baader, “Unification in Commutative Theories,” in C. Kirchner (ed.), Special Issue on Unification, J. Symbolic Computation 8, 1989.
F. Baader, “Unification in Commutative Theories, Hilbert's Basis Theorem, and Gröbner Bases,” to appear in J. ACM.
F. Baader, “Unification Theory,” these proceedings.
F. Baader, W. Nutt, “Adding Homomorphisms to Commutative/Monoidal Theories, or: How Algebra Can Help in Equational Unification,” Proceedings of the 4th International Conference on Rewriting Techniques and Applications, Springer LNCS 488, 1991.
L. Bachmair, Proof Methods for Equational Theories, Ph.D. Thesis, Dep. of Comp. Sci., University of Illinois at Urbana-Champaign, 1987.
A.P. Birjukov, “Varieties of Idempotent Semigroups,” Algebra i Logica 9, 1970; English Translation in Algebra and Logic 9, 1970.
A.H. Clifford, “The Free Completely Regular Semigroup on a Set,” J. Algebra 59, 1979.
P.M. Cohn, Universal Algebra, Harper and Row, New York, 1965.
F. Fages, G.P. Huet, “Complete Sets of Unifiers and Matchers in Equational Theories,” Proceedings of the CAAP'83, Springer LNCS 170, 1983.
C.F. Fennemore, “All Varieties of Bands I, II,” Math. Nachr. 48, 1971.
G. Grätzer, Universal Algebra, Van Nostrand, Princeton, 1968.
J.A. Gerhard, “The Lattice of Equational Classes of Idempotent Semigroups,” J. Algebra 15, 1970.
J.A. Gerhard, “Free Completely Regular Semigroups I, II,” J. Algebra 82, 1983.
J.A. Gerhard, M. Petrich “The Word Problem for Orthogroups,” Can. J. Math. 4, 1981.
A. Herold, Combination of Unification Algorithms in Equational Theories, Dissertation, Fachbereich Informatik, Universität Kaiserslautern, 1987.
H. Herrlich, G.E. Strecker, Category Theory, Allyn and Bacon, Boston, 1973.
G.P. Huet, “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems,” J. ACM 27, 1980.
J. Jaffar, J.L. Lassez, M. Maher, “A Theory of Complete Logic Programs with Equality,” J. Logic Programming 1, 1984.
J.P. Jouannaud, H. Kirchner, “Completion of a Set of Rules Modulo a Set of Equations,” SIAM J. Computing 15, 1986.
J.P. Jouannaud, C. Kirchner, “Solving Equations in Abstract Algebras: A Rule-Based Survey of Unification,” Preprint, 1990. To appear in the Festschrift to Alan Robinson's birthday.
A.J. Nevins, “A Human Oriented Logic for Automated Theorem Proving,” J. ACM 21, 1974.
W. Nutt, “Unification in Monoidal Theories,” Proceedings 10th International Conference on Automated Deduction, Springer LNCS 449, 1990.
W. Nutt, “The Unification Hierarchy is Undecidable,” to appear in J. Automated Reasoning. Also SEKI-Report SR-89-06, Universität Kaiserslautern, West Germany, 1989.
G. Peterson, M. Stickel, “Complete Sets of Reductions for Some Equational Theories,” J. ACM 28, 1981.
M. Petrich, “The Structure of Completely Regular Semigroups,” T. AMS 189, 1974.
M. Petrich, “Varieties of Orthodox Bands of Groups,” Pacific J. Math. 56, 1975.
M. Petrich, “On Varieties of Completely Regular Semigroups,” Semigroup Forum 25, 1982.
G. Plotkin, “Building in Equational Theories,” Machine Intelligence 7, 1972.
L. Polak, “On Varieties of Completely Regular Semigroups,” Semigroup Forum 32, 1985.
M. Schmidt-Schauß, “Unification under Associativity and Idempotence is of Type Nullary,” J. Automated Reasoning 2, 1986.
J.H. Siekmann, “Unification Theory: A Survey,” in C. Kirchner (ed.), Special Issue on Unification, Journal of Symbolic Computation 7, 1989.
J.R. Slagle, “Automated Theorem Proving for Theories with Simplifiers, Commutativity and Associativity,” J. ACM 21, 1974.
M.E. Stickel, “Automated Deduction by Theory Resolution,” J. Automated Reasoning 1, 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Baader, F. (1992). Unification in varieties of completely regular semigroups. In: Schulz, K.U. (eds) Word Equations and Related Topics. IWWERT 1990. Lecture Notes in Computer Science, vol 572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55124-7_9
Download citation
DOI: https://doi.org/10.1007/3-540-55124-7_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55124-9
Online ISBN: 978-3-540-46737-3
eBook Packages: Springer Book Archive