Monotone clones, residual smallness and congruence distributivity. (English) Zbl 0695.08012
Corresponding to each ordered set there is a variety, determined up to equivalence, generated by an algebra whose term operations are all the monotone operations on the ordered set. The author gives several characterisations of the finite bounded ordered sets for which the corresponding variety is congruence-distributive.
Reviewer: A.Hatvany
MSC:
| 08B10 | Congruence modularity, congruence distributivity |
| 06A06 | Partial orders, general |
| 08A40 | Operations and polynomials in algebraic structures, primal algebras |
Keywords:
clones; residual smallness; congruence distributivity; monotone operations; finite bounded ordered setsReferences:
| [1] | DOI: 10.1007/BF01187059 · Zbl 0292.08004 · doi:10.1007/BF01187059 |
| [2] | DOI: 10.1007/BF00400285 · Zbl 0613.08003 · doi:10.1007/BF00400285 |
| [3] | Rosenberg, Near unanimity orders (1985) |
| [4] | Rosenberg, Rozpravy Ceskoslovenské Akad. Ved 80 (1970) |
| [5] | DOI: 10.1007/BF01182246 · Zbl 0542.06001 · doi:10.1007/BF01182246 |
| [6] | DOI: 10.1007/BF02582962 · Zbl 0606.06002 · doi:10.1007/BF02582962 |
| [7] | DOI: 10.1007/BF01194521 · Zbl 0522.06002 · doi:10.1007/BF01194521 |
| [8] | Martynjuk, Problemy Kibernet. 3 pp 49– (1960) |
| [9] | DOI: 10.1002/malq.19780240111 · Zbl 0401.03008 · doi:10.1002/malq.19780240111 |
| [10] | Jónsson, Math. Scand. 21 pp 110– (1967) · Zbl 0167.28401 · doi:10.7146/math.scand.a-10850 |
| [11] | Freese, Commutator theory for congruence modular varieties (1987) · Zbl 0636.08001 |
| [12] | Hobby, The Structure of Finite Algebras (1988) · Zbl 0721.08001 · doi:10.1090/conm/076 |
| [13] | DOI: 10.1016/0012-365X(81)90201-6 · Zbl 0459.06002 · doi:10.1016/0012-365X(81)90201-6 |
| [14] | Hagemann, Arch. Math. (Basel) pp 234– (1979) · Zbl 0419.08001 · doi:10.1007/BF01238496 |
| [15] | Denecke, Preprimal Algebras (1982) |
| [16] | DOI: 10.1007/BF00400284 · Zbl 0614.08006 · doi:10.1007/BF00400284 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
