\({\mathbf Z}\)-continuous posets. (English) Zbl 0851.06003

Baranga, Andrei

Discrete Math. 152, No. 1-3, 33-45 (1996).

Summary: The concept of subset system on the category \({\mathbf P} {\mathbf o}\) of posets \(({\mathbf Z}\)-sets) was defined by J. B. Wright, E. G. Wagner and J. W. Thatcher [Theor. Comput. Sci. 7, 57-77 (1978; Zbl 0732.06001)]. The term \({\mathbf Z}\)-set becomes meaningful if we replace \({\mathbf Z}\) by ‘directed’, ‘chain’, ‘finite’. At the end of that paper [loc. cit.], the authors suggested an attempt to study the generalized counterpart of the term ‘continuous poset (lattice)’ obtained by replacing directed sets by \({\mathbf Z}\)-sets, \({\mathbf Z}\) being an arbitrary subset system on \({\mathbf P} {\mathbf o}\). We present here some results concerning this investigation. These results are generalized counterparts of some purely order-theoretical facts about continuous posets.

MSC:

06B35	Continuous lattices and posets, applications
06A15	Galois correspondences, closure operators (in relation to ordered sets)
68Q55	Semantics in the theory of computing

Keywords:

continuous lattice; algebraic poset; algebraic lattice; Moore family; closure operator; ideal; \({\mathbf Z}\)-inductive poset; subset systems of the category of posets; \({\mathbf Z}\)-sets; continuous posets

Citations:

Zbl 0732.06001

References:

[1]	Baranga, A., \(Z\)-topologies (I), Bull. Math. Soc. Sci. Math. R.S. Roumanie, 33, 207-215 (1989) · Zbl 0732.54001
[2]	Baranga, A., \(Z\)-topologies (II), Bull. Math. Soc. Sci. Math. R.S. Roumanie, 33, 291-300 (1989) · Zbl 0732.54002
[3]	Dilworth, R. P.; Crawley, P., Algebraic Theory of Lattices (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0494.06001
[4]	Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., A Compendium of Continuous Lattices (1980), Springer: Springer Berlin · Zbl 0452.06001
[5]	Grätzer, G., General Lattice Theory (1978), Birkhäuser: Birkhäuser Basel · Zbl 0385.06015
[6]	Hofmann, K. H., A note on Baire spaces and continuous lattices (1979), preprint
[7]	Hofmann, K. H.; Stralka, A. R., The algebraic theory of Lawson semilattices — applications of Galois connections to compact semilattices, Diss. Math., 137, 1-54 (1976) · Zbl 0359.06016
[8]	Lea, J. W., Continuous lattices and compact Lawson semilattices, Semigroup Forum, 13, 387-388 (1976/1977) · Zbl 0361.06006
[9]	Scott, D. S., Continuous Lattices, (Lectures Notes in Mathematics, Vol. 224 (1972), Springer: Springer Berlin), 97-136
[11]	Scott, D. S., Logic and programming languages, Comm. ACM, 20, 634-641 (1977) · Zbl 0355.68019
[12]	Wright, J. B.; Wagner, E. G.; Thatcher, J. W., A uniform approach to inductive posets and inductive closure, Theoret. Comput. Sci., 7, 55-77 (1978) · Zbl 0732.06001