A completion for distributive nearlattices. (English) Zbl 1455.06004

By a polarity a triple \((X,Y,R)\) is meant where \(X,Y\) are nonempty sets and \(R\) is a binary relation between \(X\) and \(Y\). For a poset \(P\), a completion of \(P\) is a pair \((L,e)\) where \(L\) is a complete lattice and \(e\) is an order embedding of \(P\) into \(L\). A collection \(F\) of upsets of \(P\) is \(standard\) if it contains all principal filters of \(P\), dually for a collection \(I\) of downsets of \(P\). A polarity is \(standard\) if it is of the form \((F,I,R)\) for standard collections \(F\) and \(I\). The authors introduce the so-called \((F,I)\)-compact and \((F,I)\)-dense polarities and the so-called \((F,I)\)-completion They prove that every distributive nearlattice can be embedded into a complete distributive lattice via an \((F,I)\)-completion and presented a connection with free distributive lattice extension. They study how an \(n\)-ary operation can be extended on a distributive nearlattice.


06A12 Semilattices
06B23 Complete lattices, completions
03G10 Logical aspects of lattices and related structures
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D05 Structure and representation theory of distributive lattices
Full Text: DOI


[1] Abbott, J., Semi-boolean algebra, Matematički Vesnik, 4, 177-198 (1967) · Zbl 0153.02704
[2] Araújo, J.; Kinyon, M., Independent axiom systems for nearlattices, Czech. Math. J., 61, 975-992 (2011) · Zbl 1249.06003
[3] Celani, S.; Calomino, I., Stone style duality for distributive nearlattices, Algebra Univ., 71, 127-153 (2014) · Zbl 1301.06030
[4] Celani, S.; Calomino, I., On homomorphic images and the free distributive lattice extension of a distributive nearlattice, Rep. Math. Log., 51, 57-73 (2016) · Zbl 1390.06008
[5] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures. Heldermann Verlag, Lemgo (2007) · Zbl 1117.06001
[6] Chajda, I.; Halaš, R., An example of a congruence distributive variety having no near-unanimity term, Acta Univ. M. Belii Ser. Math., 13, 29-31 (2006) · Zbl 1132.08002
[7] Chajda, I.; Kolařík, M., A decomposition of homomorphic images of nearlattices, Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica, 45, 43-51 (2006) · Zbl 1123.06002
[8] Chajda, I.; Kolařík, M., Ideals, congruences and annihilators on nearlattices, Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica, 46, 25-33 (2007) · Zbl 1147.06002
[9] Chajda, I.; Kolařík, M., Nearlattices, Discrete Math., 308, 4906-4913 (2008) · Zbl 1151.06004
[10] Cornish, W.; Hickman, R., Weakly distributive semilattices, Acta Math. Hung., 32, 5-16 (1978) · Zbl 0497.06005
[11] Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002) · Zbl 1002.06001
[12] Dunn, JM; Gehrke, M.; Palmigiano, A., Canonical extensions and relational completeness of some substructural logics, J. Symb. Log., 70, 713-740 (2005) · Zbl 1101.03021
[13] Gehrke, M.; Harding, J., Bounded lattice expansions, J. Algebra, 238, 345-371 (2001) · Zbl 0988.06003
[14] Gehrke, M.; Jansana, R.; Palmigiano, A., \( \Delta_1\)-completions of a poset, Order, 30, 39-64 (2013) · Zbl 1317.06002
[15] Gehrke, M.; Jónsson, B., Bounded distributive lattices with operators, Math. Jpn., 40, 207-215 (1994) · Zbl 0855.06009
[16] Gehrke, M.; Jónsson, B., Monotone bounded distributive lattice expansions, Math. Jpn., 52, 197-213 (2000) · Zbl 0972.06005
[17] Gehrke, M.; Jónsson, B., Bounded distributive lattice expansions, Math. Scand., 94, 13-45 (2004) · Zbl 1077.06008
[18] González, LJ, The logic of distributive nearlattices, Soft Comput., 22, 2797-2807 (2018) · Zbl 1398.06019
[19] Halaš, R., Subdirectly irreducible distributive nearlattices, Miskolc Math. Notes, 7, 141-146 (2006) · Zbl 1120.06003
[20] Hickman, R., Join algebras, Commun. Algebra, 8, 1653-1685 (1980) · Zbl 0436.06003
[21] Jónnson, B.; Tarski, A., Boolean algebras with operators. Part II, Am. J. Math., 74, 127-162 (1952) · Zbl 0045.31601
[22] Jónsson, B.; Tarski, A., Boolean algebras with operators. Part I, Am. J. Math., 73, 891-939 (1951) · Zbl 0045.31505
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.