Palestine Journal of Mathematics Vol. 5(1) (2016) , 154–158 © Palestine Polytechnic University-PPU 2016 IRREDUCIBLE ELEMENTS IN ADL’S B. Venkateswarlu, Ch. Santhi Sundar Raj and R. Vasu Babu Communicated by Ayman Badawi MSC 2010 Classifications: 06D99. Keywords and phrases: Distributive Lattice, Almost Distributive Lattice, ∧- irreducible , ∨- irreducible, Boolean algebra, Boolean ring. The authors thank Professor U. M. Swamy for his help in preparing this paper. Abstract. The notion of an Almost Distributive Lattice (ADL) is a common abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebras and Boolean rings. In this paper, we introduce the concepts of ∧− irreducible (strongly ∧− irreducible) and ∨− irreducible (strongly ∨− irreducible) elements in an ADL and prove certain properties of these. Unlike the case of distributive lattices, ∧− irreducibility and strongly ∧− irreducibility are not equivalent. However, it is proved here that an element in ADL is ∨− irreducible if and only if it is strongly ∨− irreducible. 1 Introduction The axiomatization of Boole’s propositional two valued logic led to the concept of Boolean alge- bra, which is a complemented distributive lattice. M. H. Stone [4] has proved that any Boolean algebra can made into a Boolean ring (a ring with unity in which every element is an idempo- tent) and vice-versa. The concept of an Almost Distributive Lattice (ADL) was introduced by U. M. Swamy and G. C. Rao [5] as a common abstraction of several lattice theoretic and ring the theoretic generalizations of Boolean algebra and Boolean rings. In this paper, we introduce the notions of ∧ and ∨ irreducibilities and strongly ∧ and ∨ irreducibilities among elements of an Almost Distributive Lattice (ADL) and prove certain properties of these. Unlike the case of distributive lattices, ∧− irreducibility and strongly ∧− irreducibility are not equivalent for the elements of an ADL. However, an element in an ADL is ∨− irreducible if and only if it is strongly ∨− irreducible. 2 Preliminaries In this section, we recall from [5] and [3] certain preliminary concepts and results concerning ADL’s . We refer to [1] and [2] for the elementary notions and results regarding partially ordered sets, lattices and distributive lattices. Let us begin with the following fundamental definition [5]. Definition 2.1. An algebra A =(A, ∧, ∨, 0) of type (2, 2, 0) is called an Almost Distributive Lattice (abbreviated as ADL) if it satisfies the following independent identities. (1). 0 ∧ a ≈ 0 (2). a ∨ 0 ≈ a (3). a ∧ (b ∨ c) ≈ (a ∧ b) ∨ (a ∧ c) (4). (a ∨ b) ∧ c ≈ (a ∧ c) ∨ (b ∧ c) (5). a ∨ (b ∧ c) ≈ (a ∨ b) ∧ (a ∨ c) (6). (a ∨ b) ∧ b ≈ b. The identities given above are independent, in the sense that no one of them is a consequence of others. The element 0 is called the zero element of the ADL. Example 2.2. Any distributive lattice bounded below is an ADL where the zero element is the smallest element .