×

New representations of algebraic domains and algebraic L-domains via closure systems. (English) Zbl 07395089

Summary: Closure systems (spaces) play an important role in characterizing certain ordered structures. In this paper, FinSet-bounded algebraic closure spaces are introduced, and then used to provide a new approach to constructing algebraic domains. Then, a special family of algebraic closure spaces, algebraic L-closure spaces, are used to represent algebraic L-domains. Next, algebraic approximate mappings are defined and serve as the appropriate morphisms between algebraic closure spaces, respectively, algebraic L-closure spaces. On the categorical level, we show that algebraic closure spaces (respectively, algebraic L-closure spaces,) each equipped with algebraic approximate mappings as morphisms, are equivalent to algebraic domains (respectively, algebraic L-domains) with Scott continuous functions as morphisms.

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Birkhoff, G., Lattice Theory (1967), Providence: American Mathematical Society, Providence · Zbl 0153.02501
[2] Caspard, N.; Monjardet, B., Some lattices of closure systems on a finite set, Discr. Math. Theor. Comput. Sci., 6, 163-190 (2004) · Zbl 1062.06005
[3] Caspard, N.; Monjardet, B., The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey, Discr. Appl. Math., 127, 241-269 (2003) · Zbl 1026.06008
[4] Davey, BA; Priestley, HA, Introduction to Lattices and Order (2002), Cambridge: Cambridge University Press, Cambridge
[5] Edelman, PH, Meet-distributive lattices and the anti-exchange closure, Algebra Univ., 10, 290-299 (1980) · Zbl 0442.06004
[6] Edelman, PH; Saks, ME, Combinatorial representation and convex dimension of convex geometries, Order, 5, 23-32 (1988) · Zbl 0659.06005
[7] Gierz, G.; Hofmann, KH; Keimel, K.; Lawson, JD; Mislove, MW; Scott, DS, Continuous Lattices and Domains (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1088.06001
[8] Guo, LK; Li, QG, The categorical equivalence between algebraic domains and F-augmented closure spaces, Order, 32, 101-116 (2015) · Zbl 1337.06003
[9] Hofmann, KH; Stralka, AR, The algebraic theory of compact Lawson semilattices: applications of Galois connections to compact semilattices, Dissert. Math., 137, 1-54 (1976) · Zbl 0359.06016
[10] Larsen, K.G., Winskel, G.: Using information systems to solve domain equations effectively. In: Kahn, G., MacQueen, D.B., Plotkin, G. (eds.) Semantics of Data Types. Lecture Notes in Computer Science, vol. 173, pp. 109-129. Springer, Berlin (1984) · Zbl 0539.68019
[11] Lei, Y.; Luo, M., Rough concept lattices and domains, Ann. Pure Appl. Logic, 159, 333-340 (2009) · Zbl 1169.06004
[12] Pfaltz, JL, Closure lattices, Discr. Math., 154, 217-236 (1996) · Zbl 0852.06002
[13] Pfaltz, JL; Jamison, RE, Closure systems and their structure, Inf. Sci., 139, 3-4, 275-286 (2001) · Zbl 0993.06004
[14] Raney, GN, Completely distributive complete lattices, Proc. Am. Math. Soc., 3, 5, 677-680 (1952) · Zbl 0049.30304
[15] Scott, D.: Continuous lattices. In: Lawvere, F.W. (ed.) Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics, vol. 274, pp. 97-136. Springer, Berlin (1972)
[16] Winskel, G.: An introduction to event structures. In: de Bakker, J.W., de Roever, W.P., Rozenberg, G. (eds.) Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. Lecture Notes in Computer Science, vol. 354, pp. 364-397. Springer, Berlin (1989)
[17] Zhao, DS, Closure spaces and completions of posets, Semigroup Forum, 90, 543-555 (2015) · Zbl 1467.06001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.