Abstract
A finite algebra \(\underline{A} = (A; F^{\underline{A}})\) is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case \(F^{\underline A}\) cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.
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* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.
** Research supported by the Thailand Research Fund.
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Denecke, K., Radelecki, S. & Ratanaprasert, C. On Constantive Simple and Order-Primal Algebras. Order 22, 301–310 (2005). https://doi.org/10.1007/s11083-005-9009-6
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DOI: https://doi.org/10.1007/s11083-005-9009-6