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algebra universalis

, Volume 41, Issue 4, pp 239–254 | Cite as

Ideals of Priestley powers of semilattices

  • J. D. Farley
  • 25 Downloads

Abstract.

Let X be a poset and Y an ordered space; X Y \((X^Y_\Sigma, X^Y_\Lambda)\) denotes the poset of continuous order-preserving maps from Y to X with the discrete (respectively, Scott, Lawson) topology. If S is a \(\lor\)-semilattice, \(S^\sigma\) its ideal semilattice, and T a bounded distributive lattice with Priestley dual space P(T), it is shown that the following isomorphisms hold: \((S^{P(T)})^\sigma \cong (S^\sigma) ^{P(T^\sigma)}_\Lambda.\) Moreover, \($(S^\sigma)^{P(T^\sigma)}_\Lambda \cong (S^\sigma) ^{P(T^\sigma)}$ if and only if $(S^\sigma)^{P(T^\sigma)}_\Lambda = (S^\sigma)^{P(T^\sigma)},\) and sufficient conditions and necessary conditions for the isomorphism to hold are obtained (both necessary and sufficient if S is a distributive \(\lor\)-semilattice).

Key Words and phrases: Function semilattice, ideal semilattice, semilattice, distributive lattice, Priestley duality, Scott topology, Lawson topology. 

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Copyright information

© Birkhäuser Verlag Basel, 1999

Authors and Affiliations

  • J. D. Farley
    • 1
  1. 1.Mathematical Institute University of Oxford 24–29 St. Giles’ Oxford OX1 3LB, United Kingdom, and Department of Mathematics, Vanderbilt University Nashville, TN 37240 U.S.A. e-mail: Farley@math.vanderbilt.eduUS

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