Geodesic
Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 41-51 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $A := (A,\rightarrow ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations $\alpha _p\colon x \mapsto (p \rightarrow x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the embedding of $A$ into this extension preserves all existing joins and certain “compatible” meets.
Let $A := (A,\rightarrow ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations $\alpha _p\colon x \mapsto (p \rightarrow x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the embedding of $A$ into this extension preserves all existing joins and certain “compatible” meets.
Classification : 03G25, 06A12, 06A15, 08A35
Keywords: adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; filter; Hilbert algebra; implicative semilattice; subtraction 
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Cīrulis, Jānis. Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 41-51. http://geodesic.mathdoc.fr/item/AUPO_2012_51_2_a3/

[1] Cīrulis, J.: Multipliers in implicative algebras. Bull. Sect. Log. (Łódź) 15 (1986), 152–158. | MR | Zbl

[2] Cīrulis, J.: Multipliers, closure endomorphisms and quasi-decompositions of a Hilbert algebra. In: Chajda et al., I. (eds) Contrib. Gen. Algebra Verlag Johannes Heyn, Klagenfurt, 2005, 25–34. | MR | Zbl

[3] Cīrulis, J.: Hilbert algebras as implicative partial semilattices. Centr. Eur. J. Math. 5 (2007), 264–279. | DOI | MR | Zbl

[4] Curry, H. B.: Foundations of Mathematical logic. McGraw-Hill, New York, 1963. | MR | Zbl

[5] Diego, A.: Sur les algèbres de Hilbert. Gauthier-Villars; Nauwelaerts, Paris; Louvain, 1966. | MR | Zbl

[6] Henkin, L.: An algebraic characterization of quantifiers. Fund. Math. 37 (1950), 63–74. | MR | Zbl

[7] Horn, A.: The separation theorem of intuitionistic propositional calculus. Journ. Symb. Logic 27 (1962), 391–399. | DOI | MR

[8] Huang, W., Liu, F.: On the adjoint semigroups of $p$-separable BCI-algebras. Semigroup Forum 58 (1999), 317–322. | DOI | MR | Zbl

[9] Huang, W., Wang, D.: Adjoint semigroups of BCI-algebras. Southeast Asian Bull. Math. 19 (1995), 95–98. | MR | Zbl

[10] Iseki, K., Tanaka, S.: An introduction in the theory of BCK-algebras. Math. Japon. 23 (1978), 1–26. | MR

[11] Karp, C. R.: Set representation theorems in implicative models. Amer. Math. Monthly 61 (1954), 523–523 (abstract).

[12] Karp, C. R.: Languages with expressions of infinite length. Univ. South. California, 1964 (Ph.D. thesis). | MR | Zbl

[13] Kondo, M.: Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup. Int. J. Math. 28 (2001), 535–543. | DOI | MR | Zbl

[14] Marsden, E. L.: Compatible elements in implicational models. J. Philos. Log. 1 (1972), 195–200. | DOI | MR

[15] Schmidt, J.: Quasi-decompositions, exact sequences, and triple sums of semigroups I. General theory. II Applications. In:Contrib. Universal Algebra Colloq. Math. Soc. Janos Bolyai (Szeged) 17 North-Holland, Amsterdam, 1977, 365–428. | MR

[16] Tsinakis, C.: Brouwerian semilattices determined by their endomorphism semigroups. Houston J. Math. 5 (1979), 427–436. | MR | Zbl

[17] Tsirulis, Ya. P.: Notes on closure endomorphisms of implicative semilattices. Latvijskij Mat. Ezhegodnik 30 (1986), 136–149 (in Russian). | MR