Semidistributivity and Whitman Property in implication zroupoids
-
Juan M. Cornejo
Abstract
In 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra A = 〈 A, →, 0 〉, where → is binary and 0 is a constant, is called an implication zroupoid (𝓘-zroupoid, for short) if A satisfies: (x → y) → z ≈ [(z′ → x) → (y → z)′]′, where x′ := x → 0, and 0″ ≈ 0. Let 𝓘 denote the variety of implication zroupoids and A ∈ 𝓘. For x, y ∈ A, let x ∧ y := (x → y′)′ and x ∨ y := (x′ ∧ y′)′. In an earlier paper, we had proved that if A ∈ 𝓘, then the algebra Amj = 〈A, ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ∈ 𝓘, the bisemigroup Amj is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 𝓜𝓔𝓙 of 𝓘, defined by the identity: x ∧ y ≈ x ∨ y, satisfies the Whitman Property. We conclude the paper with two open problems.
(Communicated by Anatolij Dvurečenskij)
Acknowledgement
The first author wants to thank the institutional support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas) and Universidad Nacional del Sur. The authors are grateful to the referees for their careful reading of an earlier version of this paper and for their helpful suggestions. The authors also wish to acknowledge that [20] was a useful tool during the research phase of this paper.
References
[1] Balbes, R.—Dwinger, P.: Distributive Lattices, University of Missouri Press, Columbia, 1974.Search in Google Scholar
[2] Bernstein, B. A.: A set of four postulates for Boolean algebras in terms of the implicative operation, Trans. Amer. Math. Soc. 36 (1934), 876–884.10.1090/S0002-9947-1934-1501773-0Search in Google Scholar
[3] Birkhoff, G.: Lattice Theory. 2nd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, 1948.Search in Google Scholar
[4] Burris, S.—c, H. P.: A Course in Universal Algebra, Springer-Verlag, New York, 1981. The free version (2012) is available online as a PDF file at math.uwaterloo.ca/$\sim$snburris.10.1007/978-1-4613-8130-3Search in Google Scholar
[5] c, J. M.—Sankappanavar, H. P.: Order in implication zroupoids, Studia Logica 104 (2016), 417–453.10.1007/s11225-015-9646-8Search in Google Scholar
[6] Cornejo, J. M.—Sankappanavar, H. P.: Semisimple varieties of implication zroupoids, Soft Computing 20 (2016), 3139–3151.10.1007/s00500-015-1950-8Search in Google Scholar
[7] Cornejo, J. M.—Sankappanavar, H. P.: On implicator groupoids, Algebra Universalis 77 (2017), 125–146.10.1007/s00012-017-0429-0Search in Google Scholar
[8] Cornejo, J. M.—Sankappanavar, H. P.: On derived algebras and subvarieties of implication zroupoids, Soft Computing 21 (2017), 6963–6982.10.1007/s00500-016-2421-6Search in Google Scholar
[9] Cornejo, J. M.—Sankappanavar, H. P.: Symmetric implication zroupoids and identities of Bol-Moufang type, Soft Computing 22 (2018), 4319–4333.10.1007/s00500-017-2869-zSearch in Google Scholar
[10] Cornejo, J. M.—Sankappanavar, H. P.: Implication zroupoids and identities of associative type, Quasigr. Relat. Syst. 26 (2018), 13–34.10.1007/s00500-017-2869-zSearch in Google Scholar
[11] Cornejo, J. M.—Sankappanavar, H. P.: Symmetric implication zroupoids and weak associative identities, Soft Computing 23 (2019), 6797–6812.10.1007/s00500-018-03701-wSearch in Google Scholar
[12] Cornejo, J. M.—Sankappanavar, H. P.: Implication zroupoids and Birkhoff systems, J. Algebr. Hyperstrucres Log. Algebr., Published Online: 20 May 2021, 12 pages.10.52547/HATEF.JAHLA.2.4.1Search in Google Scholar
[13] Cornejo, J. M.—Sankappanavar, H. P.: Varieties of Implication zroupoids I, preprint (2020).Search in Google Scholar
[14] Day, A.: Splitting lattices generate all lattices, Algebra Universalis 7 (1977), 163–169.10.1007/BF02485425Search in Google Scholar
[15] Freese, R.—Ježek, J.—Nation, J. B.: Free Lattices, Math. Surv. Monogr. 42, American Mathematical Society, 1995.10.1090/surv/042Search in Google Scholar
[16] Gusev, S. V.—Sankappanavar, H. P.—Vernikov, B. M.: Implication semigroups, Order 37 (2020), 271–277.10.1007/s11083-019-09503-5Search in Google Scholar
[17] Johnston-Thom, K. G.—Jones, P. R.: Semidistributive inverse semigroups, J. Austral. Math. Soc. 71 (2001), 37–51.10.1017/S1446788700002706Search in Google Scholar
[18] Jónsson, B.: Sublattices of a free lattice, Canad. J. Math. 13 (1961), 256–264.10.4153/CJM-1961-021-0Search in Google Scholar
[19] Jónsson, B.–KIEFER, J. E.: Finite sublattices of a free lattice, Canad. J. Math. 14 (1962), 487–497.10.4153/CJM-1962-040-1Search in Google Scholar
[20] Mccune, W.: Prover9 and Mace4, (2005–2010). http://www.cs.unm.edu/mccune/prover9/.Search in Google Scholar
[21] Papert, D.: Congruence relations in semilattices, J. Lond. Math. Soc. s1-39 (1964), 723–729.10.1112/jlms/s1-39.1.723Search in Google Scholar
[22] Płonka, J.: On distributive quasilattices, Fund. Math. 60 (1967), 91–200.10.4064/fm-60-2-191-200Search in Google Scholar
[23] Rasiowa, H.: An Algebraic Approach to Non-classical Logics, North-Holland, Amsterdam, 1974.Search in Google Scholar
[24] Sankappanavar, H. P.: De Morgan algebras: new perspectives and applications, Sci. Math. Jpn. 75 (2012), 21–50.Search in Google Scholar
[25] Shiryaev, V. M.: Semigroups with ∧-semidistributive subsemigroup lattice, Semigroup Forum 31 (1985), 47–68.10.1007/BF02572639Search in Google Scholar
[26] Whitman, P.: Free lattices, Ann. of Math. 42 (1941), 325–330.10.2307/1969001Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- Semidistributivity and Whitman Property in implication zroupoids
- Composition of binary quadratic forms over number fields
- On Z-Symmetric Rings
- On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
- Coefficient estimates for Libera type bi-close-to-convex functions
- Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
- On rapidly oscillating solutions of a nonlinear elliptic equation
- Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
- Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
- On absolute double summability methods with high indices
- On the continuity of lattice isomorphisms on C(X, I)
- New fixed point theorems for countably condensing maps with an application to functional integral inclusions
- Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
- The form of locally defined operators in waterman spaces
- Conformal vector fields on almost co-Kähler manifolds
- A certain η-parallelism on real hypersurfaces in a nonflat complex space form
- On log-bimodal alpha-power distributions with application to nickel contents and erosion data
- Univariate and bivariate extensions of the generalized exponential distributions
- Pellian equations of special type
Articles in the same Issue
- Regular Papers
- Semidistributivity and Whitman Property in implication zroupoids
- Composition of binary quadratic forms over number fields
- On Z-Symmetric Rings
- On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
- Coefficient estimates for Libera type bi-close-to-convex functions
- Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
- On rapidly oscillating solutions of a nonlinear elliptic equation
- Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
- Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
- On absolute double summability methods with high indices
- On the continuity of lattice isomorphisms on C(X, I)
- New fixed point theorems for countably condensing maps with an application to functional integral inclusions
- Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
- The form of locally defined operators in waterman spaces
- Conformal vector fields on almost co-Kähler manifolds
- A certain η-parallelism on real hypersurfaces in a nonflat complex space form
- On log-bimodal alpha-power distributions with application to nickel contents and erosion data
- Univariate and bivariate extensions of the generalized exponential distributions
- Pellian equations of special type
