Tbilisi Mathematical Journal

Effective codescent morphisms in the varieties determined by convergent term rewriting systems

Guram Samsonadze and Dali Zangurashvili

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It is shown that the elements of amalgamated free products in a variety of universal algebras have unique normal forms if the variety is represented by a confluent term rewriting system satisfying some additional requirements for its signature and rules. Applying this fact it is proved that any codescent morphism is effective in such varieties. In particular, this is the case for the variety of Mal'tsev algebras, the varieties of magmas with unit and two-sided inverses, idempotent quasigroups, unipotent quasigroups, left Steiner loops, and right Steiner loops.

Article information

Tbilisi Math. J., Volume 9, Issue 1 (2016), 49-64.

Received: 24 June 2015
Accepted: 15 December 2015
First available in Project Euclid: 12 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 08B05: Equational logic, Malʹcev (Malʹtsev) conditions
Secondary: 08B25: Products, amalgamated products, and other kinds of limits and colimits [See also 18A30] 18C20: Algebras and Kleisli categories associated with monads 68Q42: Grammars and rewriting systems

Variety of universal algebras normal form for an element of amalgamated free product confluent term rewriting system effective codescent morphism


Samsonadze, Guram; Zangurashvili, Dali. Effective codescent morphisms in the varieties determined by convergent term rewriting systems. Tbilisi Math. J. 9 (2016), no. 1, 49--64. doi:10.1515/tmj-2016-0005. https://projecteuclid.org/euclid.tbilisi/1528769041

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  • F. Baader and T. Nipkow, Term rewriting and all that, Cambridge University Press, 2006.
  • L. Bachmair, N. Dershowitz and J. Hsiang, Orderings for equational proofs, In 1st IEEE Symp. on Logic in Computer Science, 346-357. IEEE Computer Society Press, 1986.
  • J. Corbin and M. Bidoit, A rehabilitation of Robinson's unification algorithm, In R. Pavon, editor, Information Processing 83, 909-914, North-Holland, 1983.
  • N. Dershowitz, L. Marcus and A. Tarlecki, Existence, uniqueness, and construction of rewrite systems, SIAM J. Computing 17 (1988), 629–639.
  • Ph. Dwinger and F. M. Yaqub, Generalized free products of Boolean algebras with an amalgamated subalgebra, Neder. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math. 25(1963), 225–231.
  • T. Evans, On mutiplicative systems defined by generators and relations. I. Normal forms theorems, Proc. Cambridge Philos. Soc. 47(1951), 637–649.
  • J. Gallier, P. Narendran, D. Plaisted, S. Raatz and W. Snyder, An algorithm for finding canonical sets of ground rewrite rules in polinomial time, J. ACM 40 (1993), 1–16.
  • W. Gehrke, Detailed catalogue of canonical term rewriting systems generated automatically, Technical report no. 62, Research Institute for Symbolic Computation (RISC), Johannes Lepler University Linz, 1992.
  • J. Herbrand, Logical writings, Reidel, 1971.
  • G. Huet, A complete proof of correctness of the Knuth-Bendix completion procedure, J. Computer and System Sciences, 23 (1981), 11–21,
  • G. Janelidze and W. Tholen, Facets of descent, I. Appl. Categ. Struct. 2 (1994), 1–37.
  • G. Janelidze and W. Tholen, Facets of descent, III: monadic descent for rings and algebras. Appl. Categ. Structures 12(5-6) (2004), 461–477.
  • E. W. Kiss, L. Márki, P. Pröhle and W. Tholen, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness and injectivity, Studia Sci. Math. Hungarica 18 (1983), 79–141.
  • D. E. Knuth and P.B. Bendix, Simple word problems in universal algebra, In. J. Leech, editor, Computational Problems in Abstract Algebra, 263–297, Pergamon Press, 1970.
  • A. Martelli and U. Montanari, An efficient unification algorithm, ACM Trans. Programming Languages and Systems, 4(2) (1982) 258–282.
  • B. Mesablishvili, Pure morphisms of commutative rings are effective descent morphisms for modules – a new proof, Theory Appl. Categ. 7 (2000), no. 3, 38–42.
  • M.H.A. Newman, On theories with a combinatorial definition of "equivalence", Annals of Mathematics 43(2) (1942), 223–243.
  • M. S. Paterson and M. N. Wegman, Linear unification, J. Computer and System Sciences, 16 (1978), 158–167.
  • R. Socher-Ambrosius, Boolean algebra admits no convergent term rewriting system, In R. E. Book, editor, Proceedings 4th Conference on Rewriting Techniques and Applications, Como (Italy), Lecture Notes in Computer Science 488 (1991), 264–274.
  • D. Zangurashvili, The strong amalgamation property and (effective) codescent morphisms, Theory Appl. Categ. 11 (2003), n. 20, 438–449.
  • D. Zangurashvili, Effective codescent morphisms in some varieties of universal algebras, Appl. Categ. Struct. 22 (2014), 241–252.