Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
idempotent residuated lattice; chain; band
idempotent residuated lattice; chain; band
Summary:
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.
References:
[1] Stanovský, D.: Commutative idempotent residuated lattices. Czech. Math. J. 57 (2007), 191-200. DOI 10.1007/s10587-007-0055-7 | MR 2309960
[2] Galatos, N.: Minimal varieties of residuated lattices. Algebra Universal. 52 (2004), 215-239. DOI 10.1007/s00012-004-1870-4 | MR 2161651 | Zbl 1082.06011
[3] Jipsen, P., Tsinakis, C.: A survey of residuated lattices. Ordered Algebraic Structures (J. Martinez, ed.), Kluwer Academic Publishers, Dordrecht (2002), 19-56. MR 2083033 | Zbl 1070.06005
[4] Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Universal. 50 (2003), 83-106. DOI 10.1007/s00012-003-1822-4 | MR 2026830 | Zbl 1092.06012
[5] Blount, K., Tsinakis, C.: The structure of residuated lattices. Internat. J. Algebra Comput. 13 (2003), 437-461. DOI 10.1142/S0218196703001511 | MR 2022118 | Zbl 1048.06010
[6] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. GTM78, Springer (1981). MR 0648287 | Zbl 0478.08001
[7] Howie, J. M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, New series, Vol. 12, Oxford Univ Press, New York (1995). MR 1455373 | Zbl 0835.20077
Search
Partner of
