A study on 0-distributive nearlattice

Finally, we show that a 0-distributive near lattice is quasi-complemented if and only if A0 (S) (the double near lattice of annulets) is a Boolean subalgebra of A(S), where A(S) is the set of all annihilator ideals of S We call a near-lattice S a medial near-lattice if for all x,y,z e S. it is not difficult to see that S has the upper limit property and is therefore a near-lattice. A non-empty subset I of a near-lattice S is called a downward set if for x E S and y€I x

A nonempty subset I of an almost lattice S is called an ideal if it is a low set and closed with respect to an existing finite supernormal. Using this result, Rosen[54] proved that C = (C]r [c) when S is distributive. But in a non-distributive near network S, it is easy to show that C = c] n Ic ) when S is medial. It is not difficult to see that the filter F is almost a lattice S praide if and only if S - F is a praidela.

An almost lattice S is distributive if and only if every ideal is the intersection of all primitives that contain it.•.

Congruences

It is well known in lattice theory that a lattice is distributive if and only if every ideal is a class of some congruence. This also characterizes the near-lattice distribution, which is an extension of Theorem 3.1 of Cornish and Hickman [14]. Then Cornish and Hickman [14] Theorem 3.1 for every ideal I of S (9(i) is the smallest congruence containing I as a congruence class.

To prove the converse, let every ideal of S be a congruence class with respect to some congruence on S.

SemiBoolean algebra

Semi/alt/ce S is a semi-Boolean algebra if and only / are the following conditions saiis/iecL. It is not difficult to show that S is SemiBoolean if and only if it is sectionally complemented and distributive. Theorem 22, p. 76 that a distribution network is Boolean if and only if its primary ideals are disordered.

Due to Zaidur Rahrnan and Noor [67] we know that s e S is standard if and only if it is both modular and strongly distributive. An element s eS is called neutral if it is standard and (ii) for all x,y,t eS,. Here d is distributive but (tAd)v(rAaAb)=r

An element s eS is standard if and only if it is both modular and strongly distributive.

Modular ideals in a nearlattice

According to him, a network L with 0 is called a 0-distributive network if aA(hvc)=0 holds for all a,b,ceL with aAb=0=aAc. In other words, a lattice with 0 is 0-distributive if and only if for every a E L the set of elements unrelated to a is an ideal L. In fact, Jayaram [30] stated the condition of the 0-distributive almost lattice given in this chapter as a weakly 0-distributive semilattice in a general semilattice.

An ideal I of a lattice L is called a semi-prime ideal if for all X,y,ZEL, xAyel and xAzEI implies XA(yvz)EJ. Thus, for lattice L with 0, L is called 0-distributive if and only if (0] is a semi-prime ideal. Then we include a number of separation properties in a general near lattice with respect to the annihilating ideals. Let we define a 0-distributive near lattice as follows: A near lattice S with 0 is called 0- distributive if for all X,y,ES with XAy=0=XA: and yv: exists.

A proper filter M of an approximate lattice S is called maximal if for every filter Q with Q M implies either Q = M or Q = S. Since an approximate lattice S with 1 is a lattice, so the concept of pseudocomplementarity is not possible in a general network nearby. . But i(s) is pseudocomplement and the ideal J is the pseudocomplement of both (t] and (r). Again, Figure 3.1 gives an approximately nondistributive lattice S where it) is pseudocomplemented.

From Baziar Rahman [9] we know that a near-lattice S is distributive if and only if i(s) is distributive, which is also equivalent to that D(S), the lattice of filters of S is distributive. Proof, let F be a good filter in S with 0. Let 'F be the set of all good filters containing F. Conversely, if the appropriate filter M is not maximal, then if 0 € S there exists a maximal filter N such that M c N.

Let I' be a prime downward set of L and let y be the set of all prime downward sets J such that J c: P. Then P is not empty since P E X. Let C be a chain in x and let. If in a 0-distributive near lattice S o}# A is the intersection of all non-:ero-ideals of then A = e S x}' toil.

WYMNAM

Then by (v) there exists a minimal minimal ideal P such that aP. Thus aEL — J' and L-P is a main filter. Then a prime ideal P containing tyy is a minimal prime ideal containing x}1 if and only if there is p € P. Thus, for the approximate lattice S with 0, S is called 0-distributive if and only if (0] is a semisimple ideal in S.

From [68], it is known that for every subgroup A of a near network S, A1 is a semisimple ideal if S is 0-distributive. Thus, for an approximate lattice S with 0, S is called 0-distributive if and only if (0] is a semisimple ideal. The non-empty intersection of all simple ('semi-prime') ideals of an adjacent lattice is an ideal seinE-president.

Then the following conditions are equivalent. ii) Every maximum/filter of S not connected to .1 is prime. ii) Every downward minimal prime containing J is a minimal primedel containing I. Every filter not connected to J is disjoint with a minimal primedel containing J. Then by (ii) S-A is a primedel and hence A minimal praidela. ii); Let F be a maximal filter disjoint with J. Then by (v) there exists a minimal primitive P such that a 0 P, which implies a e S - P and S-P is a primitive filter.

Then a prime ideal P containing {x} is a minimal prime ideal containing {X}' if and only iffbr p E F there exists q E S - P such that p A q E. Since every distributive near lattice with 0 is 0-distributive, then is 0- distributive and so (iii) holds. (iii) (i); Let 0 be a congruence of S, where J is the zero element of the 0-distributed near lattice. It now follows from the prime separation theorem for distributive near lattices due to Baziar Rahman [9] Theorem 1.2.5, That there exists a prime ideal - of disjoint to Then clearly P =.

CHAPTER III

Some Characterizations of 0-distributive Nearlattice

If A is a nonempty subset of S and B is a proper filter that cuts A, then there is a minimal prime ideal containing . For every non-zero element a € S and every proper filter B containing a there is a prime ideal containing B and breaking from B. For every non-zero element an E S and every proper filter B that contains a, there is a prime/liter containing B and disjoint of a}'. f).

For every non-zero element a € S and every prime set B that does not contain a, there is a prime/liter containing S - B and disjoint of {a}'. So by (ii) there is a prime ideal containing {a}' and disjoint from B. iii)=(iv); This is trivial since P is a prime ideal of a near lattice S if and only if S — P is a prime filter. If A and B are filters of S such that A and B0 are disjoint, then there is a minimal prime ideal containing B0 and disjoint from A.

If A is a filter of S and B is a prime-down set containing A 0, then there is a minimal prime ideal containing A and contained in B. If A is a filter of S and B is a prime-down set containing A 0 is a prime filter containing S - B and disjoint from A0. For every nonzero element a E S and every prime subset B containing lar, there is a prime filter containing S - B and disjoint from {a}°.

Then by (ii) there is a minimal prime ideal containing A ° and disjoint from S - B, which is contained in B. For any nonempty subset A of S, A1 is the intersection of all the minimal prime ideals that do not contain A. Then there exists teEA such that ten. If Q is any minimal prime ideal of S such that QA. iEIvAj), then QAnA for some jEl.

For any filler A qf 5, A 0 is the inection of all the minimal prime ideals di/oint from A.

Annulets in a 0-distributive Nearlattice

A 0-distributive close network S is quasi-complemented if and only if for every x € S there exists a y E S such that (x]** = yT. Then S is quasi-complemented if and only if it is a sectional quasi- complemented and has an element d such that dT = to]. Conversely, suppose that S is sectionally quasi-complemented and there exists an element d ES with dr = CO].

Recall that an ideal I in an approximately distributed lattice S is called an a -ideal if for every x E 5, x € I implies x]. Furthermore, the above conditions are equivalent to quasi-complete S if and only if there exists an element d E S such that dT = o]. Then by Lemma 4.4.13, A0(s) is relatively satisfied, and so by Proposition 2.7 in [13,] by Cornish S is almost partially satisfied.

We recall that a lattice L with 0 is called 0-distributive if for all a,b,cEL with aAb=0=aAc mean aA(bvc)=0. Due to Varlet [66] we know that S with 0 is 0-modular if it does not contain any non-modular living element of the pentagonal subspace including 0. Also S with 0 is 0-distributive if it does not contain five modular elements, but non-distributive, including 0.

A lattice L with 1 dual is called semi-complemented if for every aEL (a#0) there exists bEL, b:#l, such that avh=l. A near-lattice S with 0 is said to be weakly complemented if for each individual element a,beS there exists a CES such that a Ac = 0 but bAc#0. A near lattice S with 0 is 0-modular if for all a,b,c ES with c :!~ a, aAh=O,avh=cvh implies that ac,mitedavb exists.

A near lattice S with 0 is 0-modular if and only if/he filter lattice of the interval 0, x] for every x E S is I-modular. A neariaattice S with 0 is a semi-Boolean lattice if and only if the following conditions are satisfied .. iii) FqO, xD is semi-complemented for every x E S.

Introduction

If P is a prime ideal in a 0-distributive near lattice S, then the ideal 0(P) is the intersection of all minimal prime ideals in P. A near lattice S with 0 is called a normal near lattice if each prime ideal contains a unique minimal prime ideal. Now we generalize part of his result in the case of a 0-distributive near-lattice. the following conditions are equivalent. i) Each prime ideal contains a unique minimal prime ideal, i.e. S is normal.

A near lattice S with 0 is said to be a comaximal near lattice if there are two minimal prime ideals of . Using the following example, Nag, Begum and Talukder show that this not all S-algebra is a D-algebra in [38]. To prove this, we need the following lemma, which is due to Noor and Razia Sultana [40].

A lattice L with I is 1-distributive and only if tbr any a # 1 in L there exists a prime ideal of L containing a. Any two distinct minimal initial ideals of t] are comatose. xiinal fin' each t of S. iii,) Eveiy prime ideal in t] contains a unique minimum ideal of ct]; t E S. Here S1 is 0-distributive and only the principal ideals are (a], (b] and they are actually minimal simple ideals.