## Lattice of closure endomorphisms of a Hilbert algebra.(English)Zbl 1437.03183

Summary: A closure endomorphism of a Hilbert algebra $$A$$ is a mapping that is simultaneously an endomorphism of and a closure operator on $$A$$. It is known that the set $$C E$$ of all closure endomorphisms of $$A$$ is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of $$A$$, anti-isomorphic to the lattice of certain closure retracts of $$A$$, and compactly generated. The set of compact elements of $$C E$$ coincides with the adjoint semilattice of $$A$$; conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.

### MSC:

 03G25 Other algebras related to logic 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06F35 BCK-algebras, BCI-algebras 08A35 Automorphisms and endomorphisms of algebraic structures
Full Text:

### References:

 [1] Birkhoff, G., Lattice Theory, (1967), American Mathematical Society: American Mathematical Society, Providence [2] Busneag, D.; Rudeanu, S., A glimpse of deductive systems in algebra, Cent. Eur. J. Math., 8, 688-705, (2010) · Zbl 1216.03066 [3] Celani, S. A.; Jansana, R., On the free implicative semilattice extension of a Hilbert algebra, Math. Log. Quart., 58, 188-207, (2012) · Zbl 1247.03136 [4] Chajda, I., The lattice of deductive systems of a Hilbert algebra, Southeast Asian Bull. Math., 26, 21-36, (2002) [5] Cı̄rulis, J., On closure endomorphisms of implicative semilattices, Bull. Sect. Log. (Łódź), 14, 52-56, (1985) · Zbl 0576.03039 [6] Cı̄rulis, J., Multipliers in implicative algebras, Bull. Sect. Log. (Łódź), 15, 152-158, (1986) · Zbl 0634.03067 [7] Cı̄rulis, J.; Chajda, I., Contributions to General Algebra, 16, Multipliers, closure endomorphisms and quasi-decompositions of a Hilbert algebra, 25-34, (2005), Verlag Johannes Heyn · Zbl 1082.03056 [8] Cı̄rulis, J., Hilbert algebras as implicative partial semilattices, Centr. Eur. J. Math., 5, 264-279, (2007) · Zbl 1125.03047 [9] Cı̄rulis, J., Adjoint semilattice and minimal Brouwerian extensions of a Hilbert algebra, Acta Univ. Palacki. Olomouc., Fac. rer. nat., Math., 51, 41-51, (2012) · Zbl 1280.03063 [10] Cornish, W. H., A multiplier approach to implicative BCK-algebras, Math. Seminar Notes (Kobe Univ.), 8, 157-169, (1980) · Zbl 0465.03029 [11] Curry, H. B., Foundations of Mathematical Logic, (1977), Dover Publications · Zbl 0396.03001 [12] Diego, A., Sur les Algèbres de Hilbert, (1966), Gauther-Villars/E. Nauwelaerts [13] Gaitán, H., Endomorphisms of implication algebras, Demonstratio Math., 47, 284-288, (2014) · Zbl 1309.03025 [14] Gaitán, H., Congruences and closure endomorphisms of Hilbert algebras, Commun. Algebra, 43, 1135-1145, (2015) · Zbl 1320.03090 [15] Hong, S. M.; Jun, Y. B., On deductive systems of Hilbert algebras, Comm. Korean Math. Soc., 11, 595-600, (1996) · Zbl 0946.03079 [16] Katrińák, T., Die Kennzeichnung der distributiven pseudokomplementären Halbverbände, J. Reine Angew. Math., 241, 160-179, (1970) · Zbl 0192.33503 [17] Marsden, E. L., Compatible elements in implicational models, J. Philos. Log., 1, 195-200, (1972) [18] Marsden, E. L., A note on implicative models, Notre Dame J. Formal Logic, 14, 139-140, (1973) · Zbl 0214.00804 [19] Nemitz, W. C., Implicative semi-lattices, Trans. Amer. Math. Soc., 117, 128-142, (1965) · Zbl 0128.24804 [20] Porta, H., Sur quelques algèbres de la logique, Port. Math., 40, 41-47, (1981) · Zbl 0581.03046 [21] Rasiowa, H., An Algebraic Approach to Non-Classical Logics, (1974), PWN/North Holland Publication · Zbl 0299.02069 [22] Schmidt, J., Zur Characterisiering der Kongruenzverbände der Verbände, Mat. Časopis, 18, 3-20, (1968) [23] Schmidt, J., Quasi-decompositions, exact sentences, and triple sums of semigroups, I, II, Colloq. Math. Soc. J. Bolyai, 17, 399-429, (1975) [24] Schmidt, J., Binomial pairs, semi-Brouwerian and Brouwerian semilattices, Notre Dame J. Formal Log., 19, 421-434, (1978) · Zbl 0351.02040 [25] Tsinakis, C., Brouwerian semilattices determined by their endomorphism semigroups, Houston J. Math., 5, 427-436, (1979) · Zbl 0431.06003 [26] Y. P. Tsirulis, Remarks on closure endomorphisms of implicative semilattices (Russian), Latvijskij Matematicheskij Ezhegodnik, Vol. 30 (Zinatne Publisher, 1986), pp. 136-149. [27] Xu, S., On the adjoint monoids of implicative BCK-algebras, Southeast Asian Bull. Math., 26, 535-540, (2002) · Zbl 1017.06011
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.