INMABB   05456
INSTITUTO DE MATEMATICA BAHIA BLANCA
Unidad Ejecutora - UE
artículos
Título:
Order in Implication Zroupoids
Autor/es:
JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR
Revista:
STUDIA LOGICA
Editorial:
SPRINGER
Referencias:
Año: 2016 vol. 104 p. 417 - 453
ISSN:
0039-3215
Resumen:
The variety $mathbf{I}$ of implication zroupoids (using a binary operation $o$ and a constant $0$) was defined and investigated by Sankappanavar in cite{sankappanavarMorgan2012}, as a generalization of De Morgan algebras. Also, in cite{sankappanavarMorgan2012}, several new subvarieties of $mathbf{I}$ were introduced, including the subvariety $mathbf{I_{2,0}}$, defined by the identity: $x´´ approx x$, which plays a crucial role in this paper. Some more new subvarieties of $mathbf{I}$ are studied in cite{CoSa2015aI} that includes the subvariety $mathbf{SL}$ of semilattices with a least element $0$; and an explicit description of semisimple subvarieties of $mathbf{I}$ is given in cite{CoSa2015semisimple}.It is a well known fact that there is a partial order (denote it by $sqsubseteq$) induced by the operation $land$, both in the variety $mathbf{SL}$ of semilattices with a least element and in the variety $mathbf{DM}$ of De Morgan algebras. As both $mathbf{SL}$ and $mathbf{DM}$ are subvarieties of $mathbf{I}$ and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation $sqsubseteq$ on $mathbf{I}$ is actually a partial order in some (larger) subvariety of $mathbf{I}$ that includes both $mathbf{SL}$ and $mathbf{DM}$.The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety $mathbf{I_{2,0}}$ is a maximal subvariety of $mathbf{I}$ with respect to the property that the relation $sqsubseteq$ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in $mathbf{I_{2,0}}$ that can be defined on an $n$-element chain (herein called $mathbf{I_{2,0}}$-chains), $n$ being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each $n in mathbb{N}$, there are exactly $n$ nonisomorphic $mathbf{I_{2,0}}$-chains of size $n$.