Characterizations of m - Normal Nearlattices in terms of Principal n - Ideals

A convex subnearlattice of a nearlattice S containing a fixed element n∈S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal nideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In this paper, we include several characterizations of those Pn(S) which form m-normal nearlattices. We also show that Pn(S) is m-normal if and only if for any m+1 distinct minimal prime n-ideals Po,P1,...., Pm of S, Po ∨ ... ∨ Pm = S. AMS Subject Classifications (2000): 06A12, 06A99, 06B10 Introduction Lee in [9], also see Lakser [7], has determined the lattice of all equational subclasses of the class of all pseudo-complemented distributive lattices. They are given by B-1⊂ Bo ⊂ .... ⊂ Bm ⊂ .... ⊂ Bω ,where all the inclusions are proper and Bω is the class of all pseudo-complemented distributive lattices, B-1 consists of all one element algebra, Bo is the variety of Boolean algebras while Bm, for -1 ≤ m < ω consists of all algebras satisfying the equation (x1 ∧ x2 ∧ .... ∧ xm) ∨ (x ∨ = n i 1 1 ∧ x2 ∧ ....∧ xi-1 ∧ xi ∧ xi+1 ∧ .....∧ xm) = 1 where x denotes the pseudo-complemente of x. Thus B1 consists of all Stone algebras. Davey [4] has independently given several characterizations of (sectionally) Bm and relatively Bm-lattices. On the other hand Cornish in [3] has studied M. S. Raihan : Rajshahi University J. of Sci. 38, 49-59 (2010) 50 distributive lattices (without pseudo-complementation) analogues to Bm-lattices and relatively Bm-lattices. A distributive nearlattice S with 0 is called m-normal if each prime ideal of L contains at most m-minimal prime ideals. For a fixed element n∈S, a convex subnearlattice containing n is called an n-ideal. An n-ideal generated by a finite number of elements a1, a2,....,an is called a finitely generated n-ideal, denoted by < a1, a2,....,an >n. The set of all finitely generated n-ideals is a nearlattice denoted by Fn(S). An n-ideal generated by a single element is called a principal n-ideal is denoted by Pn(S). In this paper we include several characterizations of those Pn(S) which form mnormal nearlattices. We show that Pn(S) is m-normal if and only if for any m+1 distinct minimal prime n-ideals Po,P1,...., Pm of S, Po ∨ ... ∨ Pm = S. We start the paper with the following result on n-ideals due to Latif and Noor [8]. Lemma 1.1 For a central element n∈S, Pn(S) ≅ (n] × [n). Following result is also essential for the development of this paper, which is due to Ali [1,Theorem 1.1.12]. Lemma 1.2 Let S be a distributive near-lattice with an upper element n and let I , J be two n-ideals of S. Then for any x ∈ I ∨ J, x ∨ n = i ∨ j and x ∧ n = i ∧ j for some i , i ∈ I , j ,j ∈ J with i ,j ≥ n and i , j ≤ n. Now we include the following result which is due to Noor and Ali [10] and this is a generalization of [2, Lemma 3.6]. A prime n-ideal P is said to be a minimal prime n-ideal belonging to n-ideal I if (i) I ⊆ P and (ii) There exists no prime n-ideal Q such that Q ≠ P and I ⊆ Q ⊆ P. Characterizations of m-Normal Nearlattices 51 A prime n-ideal P of a nearlattice S is called a minimal prime n-ideal if there exists no prime n-ideal Q such that Q ≠ P and Q ⊆ P. Thus a minimal prime nideal is a minimal prime n-ideal belonging to {n}. Following lemma will be needed for further development of this paper. This is [3, Lemma 3.6] and is easy to prove. So we omit the proof. The following result is [4, Lemma 2.2] which also follows from the corresponding result for commutative semi-groups due to Kist [6]. Lemma 1.3 Let M be a prime ideal containing an ideal J in a distributive medial nearlattice. Then M is a minimal prime ideal belonging to J if and only if for all x∈M, there exists x'∉M such that x ∧ x' ∈J. Now we generalize this result for n-ideals. Lemma 1.4 Let n be a medial element and M be a prime n-ideal containing an n-ideal J. Then M is a minimal prime n-ideal belonging to J if and only if for all x∈M there exists x'∉M such that m(x, n, x')∈J. Proof. Let M be a minimal prime n-ideal belonging to J and x∈M. Then by [11], < < a >n, J > ⊄ M. So there exists x’ with m(x, n, x') ∈J such that x'∉M. Conversely, suppose x∈M, then there exists x'∉M such that m(x, n, x')∈J. This implies x'∉M, but x'∈< < x >n, J >, that is < < x >n, J > ⊄ M. Hence by [10], M is a prime n-ideal belonging to J. Davey in [4, Corollary 2.3] used the following result in proving several equivalent conditions on Bm-lattices. On the other hand, Cornish in [3] has used this result in studing n-normal lattices. Proposition 1.5 Let Mo,..., Mn be n+1 distinct minimal prime ideals of a distributive nearlattice S. Then there exists ao, a1,...., an∈S such that ai ∧ aj ∈ J (i ≠ j) and aj∉Mj, j = 0, 1,..., n. Now we generalize the above result in terms of n-ideals. Proposition 1.6 Let S be a distributive nearlattice and n∈S is medial. Suppose Mo,.., Mm be m+1 distinct minimal prime n-ideals containing n-ideal J. Then M. S. Raihan : Rajshahi University J. of Sci. 38, 49-59 (2010) 52 there exists ao, a1,..., an∈S such that m(ai, n, aj) ∈J ( i ≠ j) and aj∉Mj (j = 0, 1,...., m). Proof. Let n =1. Let xo∈M1 Mo and x1∈Mo – M1. Then by Lemma 1.3, there exists x1'∉Mo such that m(xo, n, x1')∈J. Hence a1=x1, ao = m(xo, n, x1') are the required elements. Observe that m(ao, n, a1) = m(m(xo, n, x1'), n, x1) = (xo ∧ x1 ∧ x1') ∨ (xo ∧ n) ∨ (x1 ∧ n) ∨ (x1' ∧ n) = (xo ∧ m(x1, n, x1')) ∨ (xo ∧ n) ∨ (m(x1, n, x1') ∧ n) = m(xo, n, m(x1, n, x1')) Now, m(x1, n, x1') ∧ n ≤ m(xo, n, m(x1, n, x1')) ≤ m(x1, n, x1') ∨ n and m(x1, n, x1')∈J, so by convexity m(ao, n, a1)∈J. Assume that, the result is true for n = m-1, and let Mo,...,Mm be m+1 distinct minimal prime n-ideals. Let bj (j = 0, 1,...., m-1) satisfy m(bi, n, bj)∈J (i ≠ j) and bj∉Mj. Now choose bm∈Mm UM 1