# Hilbert algebras as implicative partial semilattices

Open Mathematics (2007)

- Volume: 5, Issue: 2, page 264-279
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topJānis Cīrulis. "Hilbert algebras as implicative partial semilattices." Open Mathematics 5.2 (2007): 264-279. <http://eudml.org/doc/269445>.

@article{JānisCīrulis2007,

abstract = {The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.},

author = {Jānis Cīrulis},

journal = {Open Mathematics},

keywords = {Compatible elements; filter; Hilbert algebra; implication; implicative semilattice; meet; partial semilattice; compatible elements; compatible meet operation},

language = {eng},

number = {2},

pages = {264-279},

title = {Hilbert algebras as implicative partial semilattices},

url = {http://eudml.org/doc/269445},

volume = {5},

year = {2007},

}

TY - JOUR

AU - Jānis Cīrulis

TI - Hilbert algebras as implicative partial semilattices

JO - Open Mathematics

PY - 2007

VL - 5

IS - 2

SP - 264

EP - 279

AB - The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.

LA - eng

KW - Compatible elements; filter; Hilbert algebra; implication; implicative semilattice; meet; partial semilattice; compatible elements; compatible meet operation

UR - http://eudml.org/doc/269445

ER -

## References

top- [1] J.C. Abbott: “Semi-boolean algebra”, Matem. Vestnik N. Ser., Vol. 4(19), (1967), pp. 177–198. Zbl0153.02704
- [2] J.C. Abbott: “Implication algebra”, Bull. Math. Soc. Sci. Math. R. S. Roumanie, Vol. 11, (1967), pp. 3–23.
- [3] J. Berman and W.J. Block: “Free Lukasiewicz and hoop residuation algebras”, Studia Logica, Vol. 68, (2001), pp. 1–28.
- [4] D. Busneag: “A note on deductive systems of a Hilbert algebra”, Kobe J. Math., Vol. 2, (1985), pp. 29–35. Zbl0584.06005
- [5] D. Busneag: “Hertz algebras of fractions and maximal Hertz algebras of quotients”, Math.Japon., Vol. 39, (1993), pp. 461–469. Zbl0810.06011
- [6] I. Chajda: “The lattice of deductive systems on Hilbert algebras”, Southeast Asian Bull. Math., Vol. 26, (2002), pp. 21–26. http://dx.doi.org/10.1007/s100120200022
- [7] I. Chajda and R. Halaš: “Order algebras”, Demonstr.Math., Vol. 35, (2002), pp. 1–10. Zbl1051.06005
- [8] I. Chajda and Z. Seidl: “An algebraich approach to partial lattices”, Demonstr. Math., Vol. 30, (1997), pp. 485–494. Zbl0910.06006
- [9] J. Cīrulis: “Subtractive nearsemilattices”, Proc. Latvian Acad. Sci., Vol. 52B, (1998), pp. 228–233. Zbl1027.06007
- [10] J. Cīrulis: “(H)-Hilbert algebras are not same as Hertz algebras”, Bull. Sect. Log. (Lódź), Vol. 32, (2003), pp. 107–108. Zbl1114.03311
- [11] J. Cīrulis: “Hilbert algebras as implicative partial semilattices”, Abstracts of AAA-67, Potsdam,(2004), http://at.yorku.ca/cgi-bin/amca/canj-36.
- [12] J. Cīrulis: “Multipliers,closure endomorphisms and quasi-decompositions of a Hilbert algebra”, Contrib. Gen. Algebra, Vol. 16, (2005), pp. 25–34. Zbl1082.03056
- [13] H.B. Curry: Foundations of Mathematical Logic, McGraw-Hill, New York e.a., 1963. Zbl0163.24209
- [14] A. Diego: Sobre Algebras de Hilbert, Notas de Logica Mat., Vol. 12, Inst. Mat. Univ. Nac. del Sur, Bahia Blanca, 1965.
- [15] A. Diego: Les Algébres de Hilbert, Collect. de Logique Math., Sér A, Vol., 21, Gauthier-Willar, Paris, 1966.
- [16] A.V. Figallo, G. Ramón and S. Saad: “A note on Hilbert algebras with infimum”, Mat.Contemp., Vol. 24, (2003), pp. 23–37. Zbl1082.03057
- [17] A. Figallo, Jr. and A. Ziliani: “Remarks on Hertz algebras and implicative semilattices”, Bull. Sect. Logic (Lódź), Vol. 24, 2005, pp. 37–42. Zbl1114.03312
- [18] G. Grätzer: General Lattice Theory, Akademie-Verlag, Berlin, 1978. Zbl0436.06001
- [19] R. Halaš: “Pseudocomplemented ordered sets”, Arch.Math.(Brno), Vol. 29, (1993), pp. 153–160. Zbl0801.06007
- [20] R. Halaš: “Remarks on commutative Hilbert algebras”, Math.Bohemica, Vol. 127, (2002), pp. 525–529. Zbl1008.03039
- [21] L. Henkin: “An algebraic characterization of quantifiers”, Fund. Math., Vol. 37, (1950), pp. 63–74. Zbl0041.34804
- [22] S.M. Hong and Y.B. Jun: “On a special class of Hilbert algebras”, Algebra Colloq., Vol. 3, (1996), pp. 285–288. Zbl0857.03040
- [23] A. Horn: “The separation theorem of intuitionistic propositional calculus”, J. Symb. Logic, Vol. 27, (1962), pp. 391–399. http://dx.doi.org/10.2307/2964545
- [24] K. Iseki and S. Tanaka: “An introduction in the theory of BCK-algebras”, Math. Japon., Vol. 23, (1978), pp. 1–26. Zbl0385.03051
- [25] Y.B. Jun: “Deductive systems of Hilbert algebras”, Math. Japon., Vol. 42, (1996), pp. 51–54. Zbl0844.03033
- [26] Y.B. Jun: “Commutative Hilbert algebras”, Soochow J. Math., Vol. 22, (1996), pp. 477–484. Zbl0864.03042
- [27] T. Katriňák: “Pseudokomplementäre Halbverbande”, Mat. Časopis, Vol. 18, (1968), pp. 121–143. Zbl0164.00701
- [28] M. Kondo: “Hilbert algebras are dual isomorphic to positive implicative BCK-algebras”, Math. Japon., Vol. 49, (1999), pp. 265–268. Zbl0930.06017
- [29] M. Kondo: “(H)-Hilbert algebras are same as Hertz algebras”, Math. Japon., Vol. 50, (1999), pp. 195–200. Zbl0937.03073
- [30] E.L. Marsden: “Compatible elements in implicative models”, J. Philos. Logic, Vol. 1, (1972), pp. 156–161. http://dx.doi.org/10.1007/BF00650494 Zbl0259.02046
- [31] E.L. Marsden: “A note on implicative models”, Notre Dame J. Formal Log., Vol. 14, (1973), pp. 139–144. http://dx.doi.org/10.1305/ndjfl/1093890823 Zbl0214.00804
- [32] A. Monteiro: “Axiomes independants pour les algèbres de Brouwer”, Rev. Un. Mat. Argentina, Vol. 17, (1955), pp. 149–160. Zbl0072.25004
- [33] Y.S. Pawar: “Implicative posets”, Bull. Calcutta Math. Soc., Vol. 85, (1993), pp. 381–384. Zbl0813.06001
- [34] W.C. Nemitz: “On the lattice of filters of an implicative semi-lattice”, J. Math. Mech., Vol. 18, (1969), pp. 683–688. Zbl0169.02101
- [35] H. Rasiowa: An Algebraic Approach to Non-classical Logics, PWN, North-Holland, Warszawa, 1974.
- [36] S. Rudeanu: “On relatively pseudocomplemented posets and Hilbert algebras”, An. Stiint. Univ. Iaşi, N. Ser., Ia, Suppl., Vol. 31, (1985), pp. 74–77.
- [37] A. Torrens: “On the role of the polynomial (X → Y ) → Y in some implicative algebras”, Zeitschr. Log. Grundl. Math., Vol. 34, (1988), pp. 117–122. Zbl0621.03043

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.