# Implication Zroupoids and Birkhoff Systems

Document Type : Original Article

Authors

1 Departamento de Matem'atica, Universidad Nacional del Sur, Alem 1253, Bah'ia Blanca, Argentina, INMABB - CONICET.

2 Hanamantagouda P. Sankappanavar, Department of Mathematics, State University of New York, New Paltz, New York 12561, U.S.A.

Abstract

An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal{I}$-zroupoid, for short) if $\mathbf{A}$ satisfies the identities: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$, where $x' : = x \to 0$, and $0'' \approx 0$. These algebras generalize De Morgan algebras and $\vee$-semilattices with zero. Let $\mathcal{I}$ denote the variety of implication zroupoids. The investigations into the structure of $\mathcal{I}$ and of the lattice of subvarieties of $\mathcal{I}$, begun in 2012, have continued in several papers (see the Bibliography at the end of the paper). The present paper is a sequel to that series of papers and is devoted to making further contributions to the theory of implication zroupoids. The main purpose of this paper is to prove that if $\mathbf{A}$ is an algebra in the variety $\mathcal{I}$, then the derived algebra $\mathbf{A}_{mj} := \langle A; \wedge, \vee \rangle$, where $a \land b := (a \to b')'$ and $a \lor b := (a' \land b')'$, satisfies the Birkhoff's  identity (BR): $x \land (x \lor y) \approx x \lor (x \land y)$. As a consequence, the implication  zroupoids $\mathbf A$ whose derived algebras $\mathbf{A}_{mj}$ are Birkhoff systems are characterized. Another interesting consequence of the main result is that there are bisemigroups that are not bisemilattices but satisfy the Birkhoff's identity, which leads us naturally to define the variety of "Birkhoff bisemigroups'' as bisemigroups satisfying the Birkhoff identity, as a generalization of Birkhoff systems. The paper concludes with some open problems.

Keywords

#### References

[1] R. Balbes, P.H. Dwinger, Distributive lattices, University of Missouri Press, Columbia, 1974.
[2] B.A. Bernstein, A set of four postulates for Boolean algebras in terms of the implicative operation,
Transactions of the American Mathematical Society, 36 (1934), 876{884.
[3] G. Birkho , Lattice theory, Second ed., American Mathematical Society Colloquium Publications, 25,
American Mathematical Society, Providence, R.I., 1948.
[4] R.A. Borzooei, M. A. Kologani, Results on hoops, Journal of Hyper Algebraic Structures and Logical
algebras, 1(1) (2020), 61-77.
[5] S. Burris, H.P. Sankappanavar, A course in universal algebra, Springer-Verlag, New York, 1981.
[6] D. Busneag, D. Piciu, M. Istrata, The Belluce lattice associated with a bounded BCK-algebra, Journal
of Hyper Algebraic Structures and Logical algebras, 2(1) (2021), 1-16.
[7] J.M. Cornejo, H.P. Sankappanavar, Order in implication zroupoids, Studia Logica, 104(3) (2016), 417{
453.
[8] J.M. Cornejo, H.P. Sankappanavar, Semisimple varieties of implication zroupoids, Soft Computing,
20(3) (2016), 3139{3151.
[9] J.M. Cornejo, H.P. Sankappanavar, On implicator groupoids, Algebra Universalis, 77(2) (2017), 125{
146.
[10] J.M. Cornejo, H.P. Sankappanavar, On derived algebras and subvarieties of implication zroupoids, Soft
Computing, 21(23) (2017), 6963{6982.
[11] J.M. Cornejo, H.P. Sankappanavar, Symmetric implication zroupoids and identities of Bol-Moufang
type, Soft Computing, 22 (2018), 4319{4333.
[12] J.M. Cornejo, H.P. Sankappanavar, Implication zroupoids and identities of associative type, Quasigroups
and Related Systems, 26 (2018), 13{34.
[13] J.M. Cornejo, H.P. Sankappanavar, Symmetric implication zroupoids and weak associative identities,
Soft Computing, 23(6) (2019), 6797{6812.
[14] J.M. Cornejo, H.P. Sankappanavar, Semi-distributivity and whitman property in implication Zroupoids,
Mathematica Slovaka, (2021), 10 pages.
[15] S.V. Gusev, H.P. Sankappanavar, B.M. Vernikov, Implication semigroups, Order, 37 (2020), 1{7.
[16] J. Harding, A. Romanowska, Varieties of Birkho  systems, Part I, Order, 34 (2017), 45{68.
[17] W. McCune, Prover 9 and Mace 4, (2005-2010). http://www.cs.unm.edu/mccune/prover9/
[18] J. Plonka, On distributive quasilattices, Fundamenta Mathematicae, 60 (1967), 191{200
[19] H. Rasiowa, An algebraic approach to non-classical logics, North{Holland, Amsterdam, 1974.
[20] H.P. Sankappanavar, De Morgan algebras: New perspectives and applications, Scientia Mathematicae
Japonicae, 75(1) (2012), 21{50.