On free modes

Michał Marek Stronkowski

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 4, page 561-568
  • ISSN: 0010-2628

Abstract

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We prove a theorem describing the equational theory of all modes of a fixed type. We use this result to show that a free mode with at least one basic operation of arity at least three, over a set of cardinality at least two, does not satisfy identities selected by ’A. Szendrei in Identities satisfied by convex linear forms, Algebra Universalis 12 (1981), 103–122, that hold in any subreduct of a semimodule over a commutative semiring. This gives a negative answer to the question raised by A. Romanowska: Is it true that each mode is a subreduct of some semimodule over a commutative semiring?

How to cite

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Stronkowski, Michał Marek. "On free modes." Commentationes Mathematicae Universitatis Carolinae 47.4 (2006): 561-568. <http://eudml.org/doc/249878>.

@article{Stronkowski2006,
abstract = {We prove a theorem describing the equational theory of all modes of a fixed type. We use this result to show that a free mode with at least one basic operation of arity at least three, over a set of cardinality at least two, does not satisfy identities selected by ’A. Szendrei in Identities satisfied by convex linear forms, Algebra Universalis 12 (1981), 103–122, that hold in any subreduct of a semimodule over a commutative semiring. This gives a negative answer to the question raised by A. Romanowska: Is it true that each mode is a subreduct of some semimodule over a commutative semiring?},
author = {Stronkowski, Michał Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {modes; Szendrei modes; subreducts; semimodules; equational theory; modes; Szendrei modes; subreducts; semimodules; equational theory},
language = {eng},
number = {4},
pages = {561-568},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On free modes},
url = {http://eudml.org/doc/249878},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Stronkowski, Michał Marek
TI - On free modes
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 4
SP - 561
EP - 568
AB - We prove a theorem describing the equational theory of all modes of a fixed type. We use this result to show that a free mode with at least one basic operation of arity at least three, over a set of cardinality at least two, does not satisfy identities selected by ’A. Szendrei in Identities satisfied by convex linear forms, Algebra Universalis 12 (1981), 103–122, that hold in any subreduct of a semimodule over a commutative semiring. This gives a negative answer to the question raised by A. Romanowska: Is it true that each mode is a subreduct of some semimodule over a commutative semiring?
LA - eng
KW - modes; Szendrei modes; subreducts; semimodules; equational theory; modes; Szendrei modes; subreducts; semimodules; equational theory
UR - http://eudml.org/doc/249878
ER -

References

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  1. Burris S., Sankappanavar H.P., A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer, New York-Berlin, 1981; http://www.math.uwaterloo.ca/snburris/htdocs/ualg.html. Zbl0478.08001MR0648287
  2. Dojer N., Clones of modes, Contributions to General Algebra 16 (2005), 75-84. (2005) Zbl1089.08004MR2166947
  3. Golan J.S., The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Longman Scientific & Technical, Harlow, 1992. Zbl0780.16036MR1163371
  4. Ježek J., Kepka T., Medial Groupoids, Rozpravy ČSAV 93/2, Academia, Praha, 1983. MR0734873
  5. Ježek J., Kepka T., Linear equational theories and semimodule representations, Internat. J. Algebra Comput. 8 (1998), 599-615. (1998) MR1675018
  6. Romanowska A.B., Semi-affine modes and modals, Sci. Math. Jpn. 61 (2005), 159-194. (2005) Zbl1067.08001MR2111551
  7. Romanowska A.B., Smith J.D.H., Modes, World Scientific, Singapore, 2002. Zbl1060.08009MR1932199
  8. Szendrei À., Identities satisfied by convex linear forms, Algebra Universalis 12 (1981), 103-122. (1981) Zbl0458.08006MR0608653

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