Weak relative pseudocomplements in semilattices

• Autorzy: Cirulis J.
• Języki publikacji: EN

Abstrakty

EN

Weak relative pseudocomplementation on a meet semilattice S is a partial operation * which associates with every pair (x, y) of elements, where (…) an element z (the weak pseudocomplement of x relative to y which is the greatest among elements u such that (…). The element z coincides with the pseudocomplement of x in the upper section [y) and, if S is modular, with the pseudocomplement of x relative to y A weakly relatively pseudomented semilattice is said to be extended, if it is equipped with a total binary operation extending *. We study congruence properties of the variety of such semilattices and review some of its subvarieties already described in the literature.

Słowa kluczowe

EN

arithmetical; Brouwerian semilattice; congruence distributive; congruence orderable; meet semilattice; relative pseudocomplementation; sectional pseudocomplementation

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2011
• Tom: Vol. 44, nr 4
• Strony: 651--672
• Opis fizyczny: Bibliogr. 42 poz.

Twórcy

autor

Cirulis J.

• Faculty Of Computing University Of Latvia Raina B., 19 RIga, Lv-1586, Latvia,

Bibliografia

• [1] M. Abad, J. M. Cornejo, J. P. Diaz Varela, The variety generated by semi-Heyting chains, Soft Comput. 15 (2011), 721–728.
• [2] K. Auinger, The congruence lattice of a combinatorial strict inverse semigroup, Proc. Edinb. Math. Soc. 37(2) (1993), 25–37.
• [3] W. J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences I, Algebra Universalis 15 (1982), 195–227.
• [4] W. J. Blok, P. Köhler, D. Pigozzi, On the structure of varieties with equationally definable principal congruences II, Algebra Universalis 19 (1984), 334–379.
• [5] F. S. de Boer, M. Gabbrielli, M. C. Meo, Semantics and expresive power of a timed concurrent constraint language, in: G. Smolka (ed.), Principles and practice of constraint programming–CP’97: The 3rd international conference on principles and practice of constraint programming (Schloss Hagenberg near Linz, Austria, October 1997), LNCS 1330 (1997), 47–61.
• [6] F. S. de Boer, M. Gabbrielli, M. C. Meo, A timed concurrent constraint language, Inform. and Comput. 161 (2000), 45–83.
• [7] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, 1981.
• [8] I. Chajda, An extension of relative pseudocomplementation to non distributive lattices, Acta Sci. Math. (Szeged) 69 (2003), 491–496.
• [9] I. Chajda, G. Eigenthaler, H. Länger, Congruence Classes in Universal Algebra, Heldermann Verlag, Lemgo, 2003.
• [10] I. Chajda, R. Halaš, Sectionally pseudocomplemented lattices and semilattices, in: K. P. Shum et al. (eds), Advances in algebra: Proceedings of the ICM satellite conference in algebra and related topics (Hong Kong, China, August 2002), River Edge, NJ, World Scientific (2003), 282–290.
• [11] I. Chajda, R. Halaś, J. Kühr, Semilattice Structures, Heldermann Verlag, Lemgo, 2007.
• [12] I. Chajda, S. Radeleczki, On varieties defined by pseudocomplemented nondistributive lattices, Publ. Math. (Debrecen) 63 (2003), 737–750.
• [13] I. Chajda, S. R. Radeleczki, On congruences of algebras defined on sectionally pseudocomplemented lattices, in: A. G. Pinus, K. N. Ponomarev (eds): Algebra and Model Theory 5, Collection of papers from the 5th summer school “Intermediate problems of model theory and universal algebra" (Erlogol, Russia, June 26–July 1, 2005), Novosibirsk State Technical University, 2005, 8–27.
• [14] J. Cīulis, Implication in sectionally pseudocomplemented posets, Acta Sci. Math. (Szeged) 74 (2008), 477–491.
• [15] J. Cīulis, Weak relative annihilators in posets, Bull. Sect. Logic Univ. Łódź 40 (2011), 1–12.
• [16] R. Freese, J. B. Nation, Congruence lattices of semilattices, Pacific J. Math. 49 (1973), 51–58.
• [17] R. Freese, J. B. Nation, Clarifation to ‘Congruence lattices of semilattices’, http:// www.math.hawaii.edu/_ralph/Preprints/semicorr.pdf (1997).
• [18] R. Giacobazzzi, C. Palamidessi, F. Ranzato, Weak relative pseudo-complements of closure operators, Algebra Universalis 36 (1996), 405–412.
• [19] R. Giacobazzzi, F. Ranzato, Some properties of complete congruence lattices, Algebra Universalis 40 (1998), 189–200.
• [20] G. Grätzer, General Lattice Theory, Akademie-Verlag, Berlin, 1978.
• [21] R. Halaš, J. Kühr, Subdirectly irreducible sectionally pseudocomplemented semilattices, Czechoslovak Math. J. 57(132) (2007), 725–735.
• [22] P. Idziak, K. Słomczyńska, A. Wroński, Fregean varieties, Int. J. Algebra Comput. 19 (2009), 595–645.
• [23] T. Katriňák, Notes on Stone lattices, Math. Fiz. Časopis 16 (1966), 128–142.
• [24] T. Katriňák, Characterisierung der Verallgemeinerten Stonishen Halbverbande, Mat. Časopis 19 (1969), 235–247.
• [25] T. Katriňák, Die Kennzeichnung der distributiven pseudocomplementären Halbverbände, J. Reine Angew. Math. 241 (1970), 160–179.
• [26] T. Katriňák, Congruence lattices of pseudocomplemented semilattices, Semigroup Forum 55 (1977), 1–23.
• [27] P. V. R. Murty, N. K. Murty, Some remarks on certain classes of semilattices, Internat. J. Math. and Math. Sci. 5 (1982), 21–30.
• [28] D. Papert, Congruence relations in semilattices, J. London Math. Soc. 39 (1964), 723–729.
• [29] D. Pigozzi, Fregean algebraic logic, in: H. Andreka, D. Monk et. al. (eds.), Algebraic Logic, Colloq. Math. Soc. János Bolyai, 54 North-Holland publ. C. Amsterdam e.a. (1991), 473–502.
• [30] L. Połkowski, Rough Sets: Mathematical Foundations, Physica-Verlag, Heidelberg, 2002.
• [31] F. Ranzato, Pseudocomplements of closure operators on posets, Discrete Math. 248 (2002), 143–155.
• [32] J. V. Rao, E. S. R. R. Kumar, Weakly distributive and sectionally *semilattice, Asian J. Algebra 3 (2010), 36–42.
• [33] H. Rasiova, An Algebraic Approach to Non-Classical Logics, PWN-North Holland, Warszawa-Amsterdam, 1974.
• [34] H. P. Sankappanavar, Semi-Heyting algebras: an abstraction from Heyting algebras, in: M. Abad et al. (eds.), Proc. 9th “Dr. Antonio A. R. Monteiro" congr. math. (Bahía Blanca, Argentina, May 30–June 1, 2007), Bahía Blanca: Universidad Nacional del Sur, Instituto de Matemática, 2008, 33–66.
• [35] J. Schmidt, Binomial pairs, semi-Brouwerian and Brouwerian semilattices, Notre Dame J. Formal Logic 19 (1978), 421–434.
• [36] A. Sendlewski, Nelson algebras through Heyting ones, I, Studia Logica 49 (1990), 105–126.
• [37] V. M. Shiryaev, Semigroups with semidistributive subsemigroup lattices (Russian), Dokl. Akad. Nauk BSSR 29 (1985), 300–303.
• [38] V. M. Shiryaev, Locally compact, compactly coverable groups with the semi-Brouwerian lattice of closed subgroups (Russian), Teor. Polugrupp Prilozh. 11 (1993), 78–90.
• [39] V. M. Shiryaev, Identities in semi-Brouwerian algebras (Russian), Dokl. Nats. Akad. Nauk Belarusi 42(2) (1998), 49–51.
• [40] A. Ursini, On subtractive varieties, I, Algebra Universalis 40 (1994), 204–222.
• [41] J. Varlet, Congruences dans les demi-lattis, Bull. Soc. Roy. Sci. Liege 34 (1965), 231–240.
• [42] E. N. Yakovenko, Locally compact groups with a semi-Brouwer lattice of closed subgroups, Russian Math. 43(7) (1999), 72–74, translation from Izv. Vyssh. Uchebn. Zaved. Mat. 7 (1999), 75–77.