Hilbert algebras as implicative partial semilattices

• Autorzy: Jānis Cīrulis
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Compatible elements; filter; Hilbert algebra; implication; implicative semilattice; meet; partial semilattice

Kategorie tematyczne

• 03G25: Other algebras related to logic
• 06A12: Semilattices

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2007
• Tom: 5
• Numer: 2
• Strony: 264-279

Daty

wydano

2007-06-01

online

2007-06-01

Twórcy

autor

Jānis Cīrulis

• University of Latvia

Bibliografia

• [1] J.C. Abbott: “Semi-boolean algebra”, Matem. Vestnik N. Ser., Vol. 4(19), (1967), pp. 177–198.
• [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.
• [5] D. Busneag: “Hertz algebras of fractions and maximal Hertz algebras of quotients”, Math.Japon., Vol. 39, (1993), pp. 461–469.
• [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.
• [8] I. Chajda and Z. Seidl: “An algebraich approach to partial lattices”, Demonstr. Math., Vol. 30, (1997), pp. 485–494.
• [9] J. Cīrulis: “Subtractive nearsemilattices”, Proc. Latvian Acad. Sci., Vol. 52B, (1998), pp. 228–233.
• [10] J. Cīrulis: “(H)-Hilbert algebras are not same as Hertz algebras”, Bull. Sect. Log. (Lódź), Vol. 32, (2003), pp. 107–108.
• [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.
• [13] H.B. Curry: Foundations of Mathematical Logic, McGraw-Hill, New York e.a., 1963.
• [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.
• [17] A. Figallo, Jr. and A. Ziliani: “Remarks on Hertz algebras and implicative semilattices”, Bull. Sect. Logic (Lódź), Vol. 24, 2005, pp. 37–42.
• [18] G. Grätzer: General Lattice Theory, Akademie-Verlag, Berlin, 1978.
• [19] R. Halaš: “Pseudocomplemented ordered sets”, Arch.Math.(Brno), Vol. 29, (1993), pp. 153–160.
• [20] R. Halaš: “Remarks on commutative Hilbert algebras”, Math.Bohemica, Vol. 127, (2002), pp. 525–529.
• [21] L. Henkin: “An algebraic characterization of quantifiers”, Fund. Math., Vol. 37, (1950), pp. 63–74.
• [22] S.M. Hong and Y.B. Jun: “On a special class of Hilbert algebras”, Algebra Colloq., Vol. 3, (1996), pp. 285–288.
• [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.
• [25] Y.B. Jun: “Deductive systems of Hilbert algebras”, Math. Japon., Vol. 42, (1996), pp. 51–54.
• [26] Y.B. Jun: “Commutative Hilbert algebras”, Soochow J. Math., Vol. 22, (1996), pp. 477–484.
• [27] T. Katriňák: “Pseudokomplementäre Halbverbande”, Mat. Časopis, Vol. 18, (1968), pp. 121–143.
• [28] M. Kondo: “Hilbert algebras are dual isomorphic to positive implicative BCK-algebras”, Math. Japon., Vol. 49, (1999), pp. 265–268.
• [29] M. Kondo: “(H)-Hilbert algebras are same as Hertz algebras”, Math. Japon., Vol. 50, (1999), pp. 195–200.
• [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
• [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
• [32] A. Monteiro: “Axiomes independants pour les algèbres de Brouwer”, Rev. Un. Mat. Argentina, Vol. 17, (1955), pp. 149–160.
• [33] Y.S. Pawar: “Implicative posets”, Bull. Calcutta Math. Soc., Vol. 85, (1993), pp. 381–384.
• [34] W.C. Nemitz: “On the lattice of filters of an implicative semi-lattice”, J. Math. Mech., Vol. 18, (1969), pp. 683–688.
• [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.

Identyfikatory

DOI

10.2478/s11533-007-0008-2