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Adjoint semilattice and minimal Brouwerian extensions of a Hilbert algebra. (English) Zbl 1280.03063
Let \(A:=(A,\rightarrow,1)\) be a Hilbert algebra. A closure endomorphism is a mapping \(\varphi :A\rightarrow A\) which is a closure operator and an endomorphism. For example, if \(p\in A\), the mapping \(\alpha_p:A\rightarrow A\) defined by \(\alpha_px:=p\rightarrow x\) is a closure endomorphism. For every finite subset \(P:=\{p_1,...,p_n\}\) of \(A\), the mapping \(\alpha_P:=\alpha_{p_n}\circ\dots\circ\alpha_{p_1}\) is a closure endomorphism, called finitely generated (if \(P=\emptyset\), then \(\alpha_P=\varepsilon\) is the identity mapping). The set \(\mathrm{CE}^f\) of all such mappings is closed under composition and the algebra \((\mathrm{CE}^f,\circ,\varepsilon)\) is a lower bounded join-semilattice, called the adjoint semilattice of the Hilbert algebra \(A\).
In this paper it is shown that the adjoint semilattice \(\mathrm{CE}^f\) is isomorphic to the semilattice of finitely generated filters of \(A\) and subtractive (dually implicative) and its generating set turns out to be closed under subtraction and is an order dual of \(A\). The lattice of ideals of \(\mathrm{CE}^f\) is isomorphic to the lattice of filters of \(A\). A minimal Brouwerian extension of \(A\) is shown to be dually isomorphic to the adjoint semilattice of \(A\). Embedding of \(A\) into its minimal Brouwerian extension preserves all existing joins, but in this paper the author characterizes also the preserved meets.

MSC:
03G25 Other algebras related to logic
06A12 Semilattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
08A35 Automorphisms and endomorphisms of algebraic structures
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References:
[1] Cīrulis, J.: Multipliers in implicative algebras. Bull. Sect. Log. (Łódź) 15 (1986), 152-158. · Zbl 0634.03067
[2] Cīrulis, J.: Multipliers, closure endomorphisms and quasi-decompositions of a Hilbert algebra. Chajda et al., I. (eds) Contrib. Gen. Algebra Verlag Johannes Heyn, Klagenfurt, 2005, 25-34. · Zbl 1082.03056
[3] Cīrulis, J.: Hilbert algebras as implicative partial semilattices. Centr. Eur. J. Math. 5 (2007), 264-279. · Zbl 1125.03047
[4] Curry, H. B.: Foundations of Mathematical logic. McGraw-Hill, New York, 1963. · Zbl 0163.24209
[5] Diego, A.: Sur les algèbres de Hilbert. Gauthier-Villars; Nauwelaerts, Paris; Louvain, 1966. · Zbl 0144.00105
[6] Henkin, L.: An algebraic characterization of quantifiers. Fund. Math. 37 (1950), 63-74. · Zbl 0041.34804
[7] Horn, A.: The separation theorem of intuitionistic propositional calculus. Journ. Symb. Logic 27 (1962), 391-399. · Zbl 0117.25302
[8] Huang, W., Liu, F.: On the adjoint semigroups of \(p\)-separable BCI-algebras. Semigroup Forum 58 (1999), 317-322. · Zbl 0928.06012
[9] Huang, W., Wang, D.: Adjoint semigroups of BCI-algebras. Southeast Asian Bull. Math. 19 (1995), 95-98. · Zbl 0859.06016
[10] Iseki, K., Tanaka, S.: An introduction in the theory of BCK-algebras. Math. Japon. 23 (1978), 1-26. · Zbl 0385.03051
[11] Karp, C. R.: Set representation theorems in implicative models. Amer. Math. Monthly 61 (1954), 523-523
[12] Karp, C. R.: Languages with expressions of infinite length. Univ. South. California, 1964 · Zbl 0127.00901
[13] Kondo, M.: Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup. Int. J. Math. 28 (2001), 535-543. · Zbl 1007.06014
[14] Marsden, E. L.: Compatible elements in implicational models. J. Philos. Log. 1 (1972), 195-200. · Zbl 0259.02046
[15] Schmidt, J.: Quasi-decompositions, exact sequences, and triple sums of semigroups I. General theory. II Applications.Contrib. Universal Algebra Colloq. Math. Soc. Janos Bolyai (Szeged) 17 North-Holland, Amsterdam, 1977, 365-428.
[16] Tsinakis, C.: Brouwerian semilattices determined by their endomorphism semigroups. Houston J. Math. 5 (1979), 427-436. · Zbl 0431.06003
[17] Tsirulis, Ya. P.: Notes on closure endomorphisms of implicative semilattices. Latvijskij Mat. Ezhegodnik 30 (1986), 136-149
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