Residuation in finite posets
-
Ivan Chajda
Abstract
When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.
Support of the research by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAČR), project 20-09869L, as well as by ÖAD, project CZ 02/2019, and support of the research of the first author by IGA, project PřF 2021 030, is gratefully acknowledged.
(Communicated by Mirko Navara)
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.—Fazio, D.: On residuation in paraorthomodular lattices, Soft Comput. 24 (2020), 10295–10304.10.1007/s00500-020-04699-wSearch in Google Scholar
[3] Chajda, I.—Länger, H.: Residuation in orthomodular lattices, Topol. Algebra Appl. 5 (2017), 1–5.10.1515/taa-2017-0001Search in Google Scholar
[4] Chajda, I.—Länger, H.: Orthomodular lattices can be converted into left residuated l-groupoids, Miskolc Math. Notes 18 (2017), 685–689.10.18514/MMN.2017.1730Search in Google Scholar
[5] Chajda, I.—Länger, H.: Residuated operators in complemented posets, Asian-Eur. J. Math. 11 (2018), Art. ID 1850097, 15 pp.10.1142/S1793557118500973Search in Google Scholar
[6] Chajda, I.—Länger, H.: Relatively pseudocomplemented posets, Math. Bohemica 143 (2018), 89–97.10.21136/MB.2017.0037-16Search in Google Scholar
[7] Chajda, I.—Länger, H.: Residuation in modular lattices and posets, Asian-Eur. J. Math. 12 (2019), Art. ID 1950092, 10 pp.10.1142/S179355711950092XSearch in Google Scholar
[8] Chajda, I.—Länger, H.: Left residuated operators induced by posets with a unary operation, Soft Computing 23 (2019), 11351–11356.10.1007/s00500-019-04028-wSearch in Google Scholar
[9] Chajda, I.—Länger, H.—Paseka, J.: Residuated operators and Dedekind-MacNeille completion. Algebraic Perspectives on Substructural Logics, Proc. 2018 Conf. Algebra Substructural Logics Take 6, to appear; http://arxiv.org/abs/1812.09616.Search in Google Scholar
[10] Chajda, I.—Rachůnek, J.: Forbidden configurations for distributive and modular ordered sets, Order 5 (1989), 407–423.10.1007/BF00353659Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- Quasi-decompositions and quasidirect products of Hilbert algebras
- Residuation in finite posets
- On a problem in the theory of polynomials
- Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers
- Mapping properties of the Bergman projections on elementary Reinhardt domains
- Lemniscate-like constants and infinite series
- On the Oscillation of second order nonlinear neutral delay differential equations
- Oscillation theorems for certain second-order nonlinear retarded difference equations
- De la Vallée Poussin inequality for impulsive differential equations
- Hs-Boundedness of a class of Fourier Integral Operators
- Dynamical behavior of a P-dimensional system of nonlinear difference equations
- Some inequalities for exponentially convex functions on time scales
- Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations
- The sine extended odd Fréchet-G family of distribution with applications to complete and censored data
- A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations
- Simulations of nonlinear parabolic PDEs with forcing function without linearization
- An existence level for the residual sum of squares of the power-law regression with an unknown location parameter
- A relationship between the category of chain MV-algebras and a subcategory of abelian groups
Articles in the same Issue
- Regular Papers
- Quasi-decompositions and quasidirect products of Hilbert algebras
- Residuation in finite posets
- On a problem in the theory of polynomials
- Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers
- Mapping properties of the Bergman projections on elementary Reinhardt domains
- Lemniscate-like constants and infinite series
- On the Oscillation of second order nonlinear neutral delay differential equations
- Oscillation theorems for certain second-order nonlinear retarded difference equations
- De la Vallée Poussin inequality for impulsive differential equations
- Hs-Boundedness of a class of Fourier Integral Operators
- Dynamical behavior of a P-dimensional system of nonlinear difference equations
- Some inequalities for exponentially convex functions on time scales
- Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations
- The sine extended odd Fréchet-G family of distribution with applications to complete and censored data
- A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations
- Simulations of nonlinear parabolic PDEs with forcing function without linearization
- An existence level for the residual sum of squares of the power-law regression with an unknown location parameter
- A relationship between the category of chain MV-algebras and a subcategory of abelian groups
