# On free modes

Commentationes Mathematicae Universitatis Carolinae (2006)

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

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topStronkowski, 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

top- 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
- Dojer N., Clones of modes, Contributions to General Algebra 16 (2005), 75-84. (2005) Zbl1089.08004MR2166947
- Golan J.S., The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Longman Scientific & Technical, Harlow, 1992. Zbl0780.16036MR1163371
- Ježek J., Kepka T., Medial Groupoids, Rozpravy ČSAV 93/2, Academia, Praha, 1983. MR0734873
- Ježek J., Kepka T., Linear equational theories and semimodule representations, Internat. J. Algebra Comput. 8 (1998), 599-615. (1998) MR1675018
- Romanowska A.B., Semi-affine modes and modals, Sci. Math. Jpn. 61 (2005), 159-194. (2005) Zbl1067.08001MR2111551
- Romanowska A.B., Smith J.D.H., Modes, World Scientific, Singapore, 2002. Zbl1060.08009MR1932199
- Szendrei À., Identities satisfied by convex linear forms, Algebra Universalis 12 (1981), 103-122. (1981) Zbl0458.08006MR0608653

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