Relatively residuated lattices and posets
-
Ivan Chajda
,
Jan Kühr
Abstract
It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.
Communicated by Mirko Navara
Acknowledgement
We thank the anonymous referee for his/her valuable suggestions.
References
[1] Bělohlávek, R.: Fuzzy Relational Systems. Foundations and Principles, Kluwer, New York, 2002.10.1007/978-1-4615-0633-1Search in Google Scholar
[2] Chajda, I.: An extension of relative pseudocomplementation to non-distributive lattices, Acta Sci. Math. (Szeged) 69 (2003), 491–496.Search in Google Scholar
[3] Chajda, I.–-Eigenthaler, G.–-Länger, H.: Congruence Classes in Universal Algebra, Heldermann, Lemgo, 2012.Search in Google Scholar
[4] Chajda, I.–-Halaš, R.–-Kühr, J.: Semilattice Structures, Heldermann, Lemgo, 2007.Search in Google Scholar
[5] Chajda, I.–-Kühr, J.: Join-semilattices whose sections are residuated po-monoids, Czechoslovak Math. J. 58 (2008), 1107–1127.10.1007/s10587-008-0073-0Search in Google Scholar
[6] Chajda, I.–-Länger, H.: Residuation in orthomodular lattices, Topol. Algebra Appl. 5 (2017), 1–5.10.1515/taa-2017-0001Search in Google Scholar
[7] Chajda, I.–-Länger, H.: Residuated operators in complemented posets, Asian-European J. Math. 11 (2018), Art. ID 1850097.10.1142/S1793557118500973Search in Google Scholar
[8] Chajda, I.–-Länger, H.–-Paseka, J.: Residuated operators and Dedekind-MacNeille completion, Trends in Logic, Proc. AsubL - take 6 (to appear). http://arxiv.org/abs/1812.0961610.1007/978-3-030-52163-9_5Search in Google Scholar
[9] Dilworth, R. P.: Non-commutative residuated lattices, Trans. Amer. Math. Soc. 46 (1939), 426–444.10.1007/978-1-4899-3558-8_33Search in Google Scholar
[10] Galatos, N.–-Jipsen, P.–-Kowalski, T.–-Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Amsterdam, 2007.Search in Google Scholar
[11] Jipsen, P.–-Tsinakis, C.: A survey of residuated lattices. In: Ordered Algebraic Structures (J. Martínez, ed.), Kluwer, Dordrecht, 2002, pp. 19–56.10.1007/978-1-4757-3627-4_3Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Relatively residuated lattices and posets
- Strongly s-dense injective hull and Banaschewski’s theorems for acts
- Wild sets in global function fields
- Generators and integral points on elliptic curves associated with simplest quartic fields
- New Filbert and Lilbert matrices with asymmetric entries
- Returning functions with closed graph are continuous
- On sets of points of approximate continuity and ϱ-upper continuity
- Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
- On solvability of some nonlocal boundary value problems for biharmonic equation
- The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
- Strongly increasing solutions of higher-order quasilinear ordinary differential equations
- Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
- Filippov solutions of vector Dirichlet problems
- Sequences of positive homoclinic solutions to difference equations with variable exponent
- Modified Lupaş-Jain operators
- New fixed point results in bv(s)-metric spaces
- Improved Young and Heinz operator inequalities for unitarily invariant norms
- Scrutiny of some fixed point results by S-operators without triangular inequality
- The lattices of families of regular sets in topological spaces
- Iterated partial summations applied to finite-support discrete distributions
- Hamiltonicity of a class of toroidal graphs
