On Ideals of Implication Groupoids

Ideals of implication groupoids are considered. Given a subset of a distributive implication groupoid, the smallest ideal containing it is constructed. A characterization of ideals in distributive implication groupoid using upper sets is given. 1. Introduction In 50-ties L-Henkin and T-Skolem introduced the notion of Hilbert algebra as an algebraic counterpart of intuitionistic logic. A Hilbert algebra [1] is an algebra of type satisfying the axioms: ？ , ？ , ？ and imply .One can easily show that can be replaced by two rather simpler axioms: ？ (left distributivity), ？ (exchange).Chajda and Hala？ [2] introduced the concept of distributive implication groupoid and studied deductive systems, ideals, and congruence relations in distributive implication groupoid. In this paper we consider ideals in distributive implication groupoid. Given a subset of a distributive implication groupoid, we make the smallest ideal containing it. We provide an equivalent condition of the ideals using the notion of upper sets. 2. Preliminaries Definition 2.1 (see [2]). An algebra of type is called an implication groupoid if it satisfies the identities:(1) , (2) for all . Example 2.2. Let in which is defined by Then is an implication groupoid. Example 2.3. Let in which is defined by Then is an implication groupoid. Definition 2.4 (see [2]). An implication groupoid of type is called a distributive implication groupoid if it satisfies the following identity: for all . Example 2.5. Let in which is defined by Then is a distributive implication groupoid. In every implication groupoid, one can introduce the so-called induced relation by the setting Lemma 2.6 (see [2]). Let be a distributive implication groupoid. Then satisfies the identities Moreover, the induced relation is a quasiorder on , and the following relationships are satisfied:(i) ,？(ii) (iii) , (iv) implies ,(v) , (vi) implies ,(vii) , (viii) . 3. On Ideals of Implication Groupoids In this section, we study some properties of ideals in a distributive implication groupoid and give the smallest ideal containing a subset of a distributive implication groupoid. We characterize ideals in terms of upper sets. Definition 3.1 (see [2]). Let be an implication groupoid. A subset is called an ideal of if ？ , ？ imply , ？ imply . Remark 3.2. If is an ideal of an implication groupoid and , then . Definition 3.3 (see [2]). Let be an implication groupoid. A subset is called a deductive system of if ？ , ？ and imply . Lemma 3.4 (see [2]). Let be an implication groupoid. Then every ideal of is a deductive system of . Converse of the above