The structure of idempotent residuated chains

Wei Chen; Xian Zhong Zhao

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 2, page 453-479
  • ISSN: 0011-4642

Abstract

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In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation 𝒟 on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.

How to cite

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Chen, Wei, and Zhao, Xian Zhong. "The structure of idempotent residuated chains." Czechoslovak Mathematical Journal 59.2 (2009): 453-479. <http://eudml.org/doc/37934>.

@article{Chen2009,
abstract = {In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation $\mathcal \{D\}$ on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.},
author = {Chen, Wei, Zhao, Xian Zhong},
journal = {Czechoslovak Mathematical Journal},
keywords = {idempotent residuated lattice; chain; band; idempotent residuated lattice; chain; band},
language = {eng},
number = {2},
pages = {453-479},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The structure of idempotent residuated chains},
url = {http://eudml.org/doc/37934},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Chen, Wei
AU - Zhao, Xian Zhong
TI - The structure of idempotent residuated chains
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 2
SP - 453
EP - 479
AB - In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation $\mathcal {D}$ on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.
LA - eng
KW - idempotent residuated lattice; chain; band; idempotent residuated lattice; chain; band
UR - http://eudml.org/doc/37934
ER -

References

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  1. Stanovský, D., 10.1007/s10587-007-0055-7, Czech. Math. J. 57 (2007), 191-200. (2007) MR2309960DOI10.1007/s10587-007-0055-7
  2. Galatos, N., 10.1007/s00012-004-1870-4, Algebra Universal. 52 (2004), 215-239. (2004) Zbl1082.06011MR2161651DOI10.1007/s00012-004-1870-4
  3. Jipsen, P., Tsinakis, C., A survey of residuated lattices, Ordered Algebraic Structures (J. Martinez, ed.), Kluwer Academic Publishers, Dordrecht (2002), 19-56. (2002) Zbl1070.06005MR2083033
  4. Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C., 10.1007/s00012-003-1822-4, Algebra Universal. 50 (2003), 83-106. (2003) Zbl1092.06012MR2026830DOI10.1007/s00012-003-1822-4
  5. Blount, K., Tsinakis, C., 10.1142/S0218196703001511, Internat. J. Algebra Comput. 13 (2003), 437-461. (2003) Zbl1048.06010MR2022118DOI10.1142/S0218196703001511
  6. Burris, S., Sankappanavar, H. P., A Course in Universal Algebra, GTM78, Springer (1981). (1981) Zbl0478.08001MR0648287
  7. Howie, J. M., Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New series, Vol. 12, Oxford Univ Press, New York (1995). (1995) Zbl0835.20077MR1455373

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