Conditions that a strongly atomic algebraic lattice is semimodular 1

1 Introduction

The structure of a semimodular lattice plays an important role in lattice theory (see e.g. [8, 9]). For example, Grätzer and Kiss showed that each finite semimodular lattice has a cover-preserving embedding into a finite geometric lattice (see [6]), further, Czédli and Tamás Schmidt proved that each finite semimodular lattice of finite length has a cover-preserving embedding into a finite geometric lattice of the same length (see [4]). Leclerc [8] gave a necessary and sufficient condition that a finite lattice is semimodular (see also [9, 10]), Powers then generalized Leclerc’s results to posets (see [11]).

In a classic book [3] a theory of semimodular lattices in strongly atomic, algebraic lattices was developed. It was shown that if a strongly atomic, algebraic lattice L has the property that, for all a, b ∈ L, a, b ≻ a ∧ b imply a ∨ b ≻ a, b, then L is semimodular (see Theorem 3.7 of [3], p.25). In view of above theorem, every locally modular, strongly atomic, algebraic lattice is semimodular. This result reveals a local characterization relating to a semimodular lattice. However, its converse is not true (see Fig.4-1 in [3], p.31). On the other hand, it is well known that the locally distributive property and the locally modular property are applying for describing those algebraic strongly atomic lattices with unique irredundant decompositions and those with replaceable irredundant decompositions, respectively (see [3]). This leads us to further investigate the structures of a semimodular lattice from its local characterizations. In this paper, we shall focus on this problem.

In this paper, we shall define a WAGC lattice, a weakly locally lattice, a weakly locally modular lattice and a weakly locally semimodular lattice, and then characterize the semimodularity in strongly atomic algebraic lattices by terms of WAGC sublattices.

Throughout this paper the notations and terminologies of [1, 7] are used. Lattice join, meet, inclusion, proper inclusion, not inclusion, covering and not covering are denoted respectively by symbols ∨, ∧ , ≤ , < , ≰ , ≻ and ⊁. The corresponding set operations are denoted respectively by the symbols ∪, ∩ , ⊆ , ⊊ and ⊈. If S and T are sets, then S - T = {x ∣ x ∈ S, x ∉ T}. If a, b are elements of a lattice L with b ≥ a, then the quotient sublattice [a, b] is defined by [a, b] = {x ∈ L ∣ a ≤ x ≤ b}. A lattice L is strongly atomic if when ever b > a in L there is an element p ∈ L satisfying b ≥ p ≻ a. In a strongly atomic, algebraic lattice L, for each a ∈ L, let u _a denote the join of those elements covering a, that is u _a  = ∨ {p ∣ p ≻ a}. We define an atom of lattice L to be p ∈ L such that p ≻ 0 where the least element of L is 0, and we will denote the set of all atoms of L as atoms (L). For the purposes of convenience, we denote atoms (a) = {p ∣ p ∈ atoms (L) , p ≤ a}. A lattice L is semimodular if x ≻ x ∧ y implies x ∨ y ≻ y for all x, y ∈ L. A lattice L is lower semimodular if x ∨ y ≻ x implies y ≻ x ∧ y for all x, y ∈ L. Let L be a lattice, and K a sublattice of L. K is a cover-preserving sublattice of L if, for any a, b ∈ K, a ≻ b in K implies a ≻ b in L. We know from Crawely and Dilworth (Theorem 3.6 of [3]) that if an algebraic strongly atomic lattice is both semimodular and lower semimodular, then it is modular.

Let A be a set and < be a relation on A. Then (A, <) is a strict linear ordering if and only if it is transitive, irreflexive and total. Again, (A, <) is a wellordering if and only if it is a strict linear ordering and, for every non-empty S ⊆ A, S has a least element. Let (A, <) be a wellordering and let P (x) be a statement about a variable x. Suppose that, for every y ∈ A, ∀ x < y , if P ( x ) holds ⇒ P ( y ) holds .

Then, for every y ∈ A, P (y) holds (Theorem 3.6 of [12]).

2 Necessary and sufficient conditions that a strongly atomic algebraic lattice is semimodular

In this section, we first introduce the concept of a WAGC lattice, then give a necessary and sufficient condition that a strongly atomic algebraic lattice L is semimodular by using the WAGC sublattice of L.

Definition 2.1. Let L be a complete lattice with the greatest element 1 = ⋁ atoms (L), and G ( L ) = { x ∈ L ∣ x = ⋁ T , T ⊆ atoms ( L ) } .(1)

When G (L) is a sublattice of L, say L is a weakly atomic generated complete lattice (briefly: L is a WAGC lattice).

It is not true that every complete lattice with the greatest element 1 = ⋁ atoms (L) is WAGC. Fig. 2-1 is a complete lattice L in which the greatest element is a join of atoms but G (L) is not a sublattice of L since c = a ∧ b ∉ G (L) for a, b ∈ G (L).

OPEN IN VIEWER

Fig.2-1

A non-WAGC lattice.

Definition 2.2. A lattice L is weakly locally if the sublattice [a, u _a ] is WAGC for each a ∈ L. When a weakly locally lattice L further satisfies that the sublattice G ([a, u _a ]) is modular (resp. semimodular) for each a ∈ L, we refer to L as a weakly locally modular (resp. semimodular) lattice. When a weakly locally modular (resp. semimodular) lattice L further satisfies that the sublattice G ([a, u _a ]) is a cover-preserving sublattice of L for each a ∈ L, we refer to L as a cover-preserving weakly locally modular (resp. semimodular) lattice.

By Definition 2.2 it is easy to see that every weakly locally modular lattice is weakly locally semimodular.

Fig. 2-2 is a cover-preserving weakly locally modular lattice and it is also a WAGC lattice.

OPEN IN VIEWER

Fig.2-2

A cover-preserving weakly locally modular.

Theorem 2.1. If a complete strongly atomic semimodular lattice L with the greatest element 1 = ⋁ atoms (L) satisfies the following condition, then L is WAGC.

(∗) Let x, y, c ∈ G (L) and t ∈ L. If c = ⋁ atoms (x) ∩ atoms (y) ≺ t ≤ x ∧ y then there exists an element z ∈ G (L) with t ≤ z which satisfies the condition (C):

(C) For any g ∈ G (L) satisfying g ≤ z, if c ∨ g ≺ t ∨ g then atoms (c ∨ g) = atoms (t ∨ g).

Proof. With the definition of WAGC lattice, it suffices to prove that G (L) is a sublattice of L. The proof is made in two steps as follows.

G (L) is a non-empty subset and x ∨ y ∈ G (L) whenever x, y ∈ G (L).

Since 0 =⋁ ∅ and 1 = ⋁ atoms (L), 0, 1 ∈ G (L) by the definition of G (L), which implies that G (L) is non-empty. Let x, y ∈ G (L). Then by the formula (1) we have that x = ⋁ atoms (x) and y = ⋁ atoms (y). Therefore x ∨ y = ⋁ atoms (x) ∪ atoms (y) ∈ G (L).

x ∧ y ∈ G (L) whenever x, y ∈ G (L).

The proof of Statement B is divided into three aspects, as follows.

If 0 ∈ {x, y}, say x = 0. Then x ∧ y = 0 ∧ y = 0 ∈ G (L).

Case (i) atoms (x)∩ atoms (y) = ∅.

We claim that x ∧ y = 0. Suppose x ∧ y ≠ 0. The condition that L is a strongly atomic lattice implies that atoms (x∧ y) ≠ ∅, and hence that there exists an atom a ∈ atoms (x ∧ y) which implies a ≤ x and a ≤ y, i.e., a∈ atoms (x) ∩ atoms (y) = ∅, a contradiction.

Case (ii) atoms (x) and atoms (y) contain each other.

Set atoms (x) ⊆ atoms (y). Then x ≤ y and x ∧ y = x ∈ G (L).

Case (iii) atoms (x) and atoms (y) do not contain each other, and atoms (x)∩ atoms (y) ≠ ∅.

Set c = ⋁ atoms (x) ∩ atoms (y). It is easy to see that c ≤ x and c ≤ y since x = ⋁ atoms (x) and y = ⋁ atoms (y). Thus c ≤ x ∧ y.

In what follows, we shall prove that atoms ( c ) = atoms ( x ∧ y ) .(2)

In fact, if r ∈ atoms (x) ∩ atoms (y), then r ≤ x and r ≤ y. Thus r ≤ x ∧ y which implies r ∈ atoms (x ∧ y). Hence atoms (x ∧ y) ⊇ atoms (x) ∩ atoms (y). On the other hand, if r ∈ atoms (x ∧ y) then both r ≤ x and r ≤ y, and this condition implies that r ∈ atoms (x) and r ∈ atoms (y), i.e., r ∈ atoms (x) ∩ atoms (y). Thus atoms (x ∧ y) ⊆ atoms (x) ∩ atoms (y). Therefore, atoms ( x ∧ y ) = atoms ( x ) ∩ atoms ( y ) .(3)

Clearly, atoms (c) ⊆ atoms (x ∧ y) since c ≤ x ∧ y, and c = ⋁ atoms (x) ∩ atoms (y) implies atoms (c) ⊇ atoms (x) ∩ atoms (y). Thus atoms (c) = atoms (x ∧ y) by Equation (3).

Next, we shall prove c = x ∧ y. We just need to prove that c < x ∧ y will deduce a contradiction since c ≤ x ∧ y.

Suppose that c < x ∧ y. Then there exists an element t ∈ L such that c ≺ t ≤ x ∧ y since L is strongly atomic. Since L satisfies (∗), there exists an element z ∈ G (L) with t ≤ z satisfies condition (C).

Using (2) and c ≺ t ≤ x ∧ y, we know that atoms ( t ) = atoms ( c ) ,(4) which together with c ≺ t implies that t ≠ ⋁ atoms (t). Thus t ∉ G (L). The conditions t ≤ z, t ∉ G (L) and z ∈ G (L) imply that t < z. Clearly, atoms (c) ⊊ atoms (z) since c, z ∈ G (L) and z > c. Thus there exists an atom b₁ ∈ atoms (z) - atoms (c) and which implies that b₁ ≻ b₁ ∧ c = 0. Using the semimodularity of lattice L, we have that b₁ ∨ c ≻ c. It is easy to see that b₁ ∨ c ∈ G (L), which together with t ∉ G (L) implies that t ≠ b₁ ∨ c. Thus the conditions t ≻ c and b₁ ∨ c ≻ c imply that t ∧ (b₁ ∨ c) = c ≺ t. Hence, b 1 ∨ t = ( b 1 ∨ c ) ∨ t ≻ b 1 ∨ c(5) since L is semimodular.

Arrange the elements in atoms (z) - atoms (c) in a (possibly transfinite) sequence b₁, b₂, ⋯ , b_β, ⋯ (β ≤ α), where α is the least ordinal number with ∣ α ∣ = ∣ atoms ( z ) - atoms ( c ) ∣ .(6)

Next, let A = {S _ξ  ∣ ξ ≤ α} where S _ξ  = {b _η  ∣ η ≤ ξ}. Clearly, atoms ( z ) - atoms ( c ) = S α ∈ A .(7)

Suppose that γ ≤ α. Assuming that c ∨ ⋁ S ɛ ≺ t ∨ ⋁ S ɛ(8) for any ɛ < γ. We note that ⋁S _ɛ  ∈ G (L) and ⋁S _ɛ  ≤ z since S _ɛ  ⊆ atoms (z) - atoms (c). Thus atoms ( c ∨ ⋁ S ɛ ) = atoms ( t ∨ ⋁ S ɛ )(9) by condition (C).

Set S _γ  - {b _γ } = S. Then S ∈ { S λ ∣ λ < γ } .(10)

Using formula (8), we have that c ∨ ⋁ S ≺ t ∨ ⋁ S .(11)

If b _γ  ∈ atoms (c ∨ ⋁ S), then c ∨ ⋁ S = c ∨ ⋁ S _γ . Moreover, we have that b _γ  ∈ atoms (t ∨ ⋁ S) by formulas (9) and (10), and which implies that t ∨ ⋁ S = t ∨ ⋁ S _γ . Therefore c ∨ ⋁ S γ ≺ t ∨ ⋁ S γ(12) by (11).

If b _γ  ∉ atoms (c ∨ ⋁ S), then b _γ  ≻ b _γ  ∧ (c ∨ ⋁ S) =0. Thus b γ ∨ ( c ∨ ⋁ S ) ≻ c ∨ ⋁ S(13) by the semimodularity of lattice L. On the other hand, we have that t ∨ ⋁ S ∉ G (L) from formulas (8), (9) and (10). Then, it is easy to see that b _γ  ∨ (c ∨ ⋁ S) ≠ t ∨ ⋁ S since b _γ  ∨ (c ∨ ⋁ S) ∈ G (L), which together with formulas (11) and (13) yields that t ∨ ⋁ S ≻ [b _γ  ∨ (c ∨ ⋁ S)] ∧ (t ∨ ⋁ S) = c ∨ ⋁ S. Hence, we have that b _γ  ∨ t ∨ ⋁ S = (b _γ  ∨ c ∨ ⋁ S) ∨ (t ∨ ⋁ S), and (t ∨ ⋁ S) ∨ (b _γ  ∨ c ∨ ⋁ S) ≻ b _γ  ∨ c ∨ ⋁ S by the semimodularity of L. Thus b _γ  ∨ t ∨ ⋁ S ≻ b _γ  ∨ c ∨ ⋁ S. We note that b _γ  ∨ ⋁ S = ⋁ S _γ since S _γ  - {b _γ } = S. Therefore, t ∨ ⋁ S γ ≻ c ∨ ⋁ S γ .(14)

It is clear that (A, ⊆) is a wellordering. Thus, by Theorem 3.6 of [12] and formulas (8), (12) and (14), we can verify that t ∨ ⋁ S _γ  ≻ c ∨ ⋁ S _γ for any γ ≤ α. From (7), we have that t ∨ ⋁ atoms ( z ) - atoms ( c ) ≻ c ∨ ⋁ atoms ( z ) - atoms ( c ) .

Clearly, z = c ∨ ⋁ atoms (z) - atoms (c). Thusz ≺ t ∨ ⋁ atoms (z) - atoms (c). However, t ∨ ⋁ atoms (z) - atoms (c) ≤ z since z > t, a contradiction.

Therefore, c = x ∧ y and x ∧ y ∈ G (L).

This completes the proof of Statement B. □

Notice that the condition (∗) in Theorem 2.1 can not be removed generally, for instance, Fig. 2-3 is a strongly atomic semimodular lattice L with the greatest element 1 = ⋁ atoms (L). It is easy to see that L does not satisfy the condition(∗), and G (L) is not a sublattice of L since x ∧ y = t ∉ G (L) for x, y ∈ G (L). Thus the lattice L is not WAGC.

OPEN IN VIEWER

Fig.2-3

A non-WAGC semimodular lattice L.

Theorem 2.2. Every complete strongly atomic distributive lattice L with the greatest element 1 = ⋁ atoms (L) is WAGC.

Proof. With Theorem 2.1 established, it suffices to prove that the lattice L satisfies the property (∗) since a distributive lattice L is semimodular.

Set c = ⋁ atoms (x) ∩ atoms (y) ≺ t ≤ x ∧ y where x, y, c ∈ G (L) and t ∈ L. We shall prove that there exists an element z ∈ G (L) with t ≤ z satisfying the condition (C) in Theorem 2.1.

By the proof of Theorem 2.1 (that is, the proofs of formulas (2) and (4)), we know that atoms ( c ) = atoms ( t ) = atoms ( x ∧ y ) .(15)

Since 1 ∈ {x ∈ G (L) ∣ x ≥ t}, we have that {x∈ G (L) ∣ x ≥ t} ≠ ∅.

Now, let z ∈ G (L) with z ≥ t. In what follows, we shall prove that atoms ( c ∨ g ) = atoms ( t ∨ g )(16) for any g ∈ G (L) with g ≤ z. We distinguish two cases as follows.

Case (1). If g ≤ c then c ∨ g = c and t = t ∨ g since g ≤ c ≺ t. Thus by formula (15) we have atoms (c ∨ g) = atoms (t ∨ g).

Case (2). If g≰c, then there exists an atom a₁ ∈ atoms (g) - atoms (c) since c, g ∈ G (L). Thus a₁ ≻ a₁ ∧ c = 0, this implies that a 1 ∨ c ≻ c(17) by the semimodularity of L. We note that t ∉ G (L) by c ≺ t and formula (15). Thus a₁ ∨ c ∈ G (L) results that t ≠ a₁ ∨ c, which together with t ≻ c and formula (17) deduces that t ≻ t ∧ (a₁ ∨ c) = c. Therefore, a 1 ∨ t = ( a 1 ∨ c ) ∨ t ≻ a 1 ∨ c(18) since L is semimodular.

Below, we shall prove atoms ( a 1 ∨ c ) = atoms ( a 1 ∨ t ) .(19)

In fact, if atoms (a₁ ∨ c) ≠ atoms (a₁ ∨ t), then there exists an atom b₁ ∈ atoms (a₁ ∨ t) - atoms (a₁ ∨ c). Then b₁ ∉ atoms (c) = atoms (t). It follows that b₁ ∨ c ≠ t. From b₁ ∉ atoms (c) we have b₁ ≻ b₁ ∧ c = 0. Thus b 1 ∨ c ≻ c(20) by the semimodularity of L, which together with b₁ ∨ c ≠ t and t ≻ c implies that t ≻ (b₁ ∨ c) ∧ t = c. Hence, b 1 ∨ t = ( b 1 ∨ c ) ∨ t ≻ b 1 ∨ c(21) since L is semimodular.

On the other hand, we have that b₁ ∨ c ≠ a₁ ∨ c since b₁ ∉ atoms (a₁ ∨ c). Thus from formulas (17) and (20) we have a 1 ∨ b 1 ∨ c ≻ a 1 ∨ c(22) and a 1 ∨ b 1 ∨ c ≻ b 1 ∨ c(23) since L is semimodular. Again, from b₁ ∈ atoms (a₁ ∨ t) - atoms (a₁ ∨ c) we have b₁ ≤ a₁ ∨ t, thus a₁ ∨ t ≥ a₁ ∨ b₁ ∨ c since t > c. Therefore, from (17) and (20) we have a₁ ∨ b₁ ∨ c = a₁ ∨ t, which follows b₁ ∨ t ≤ a₁ ∨ t. Thus b₁ ∨ c ∨ t ≤ a₁ ∨ c ∨ t = a₁ ∨ b₁ ∨ c, which together with (21) and (23) implies that b₁ ∨ t = a₁ ∨ b₁ ∨ c. Consequently, the lattice L contains a sublattice L₀ as Fig. 2-4, contrary to fact that L is distributive by Theorem 3.3 of [3]. Therefore, the formula (19) is true.

OPEN IN VIEWER

Fig.2-4

The sublattice L₀.

Arrange the elements in atoms (g) - atoms (c) in a (possibly transfinite) sequence a₁, a₂, ⋯ , a_β, ⋯ (β ≤ α), where α is the least ordinal number with ∣ α ∣ = ∣ atoms ( g ) - atoms ( c ) ∣ .(24)

Next, let A = {S _ξ  ∣ ξ ≤ α} where S _ξ  = {a _η  ∣ η ≤ ξ}. Clearly, atoms ( g ) - atoms ( c ) = S α ∈ A .(25)

Suppose that γ ≤ α. Assuming that c ∨ ⋁ S ɛ ≺ t ∨ ⋁ S ɛ(26) and atoms ( c ∨ ⋁ S ɛ ) = atoms ( t ∨ ⋁ S ɛ )(27) for any ɛ < γ.

Set S _γ  - {b _γ } = S. Then S ∈ { S λ ∣ λ < γ } .(28)

Using formulas (26) and (27), we have that c ∨ ⋁ S ≺ t ∨ ⋁ S(29) and atoms ( c ∨ ⋁ S ) = atoms ( t ∨ ⋁ S ) .(30)

If a _γ  ∉ atoms (c ∨ ⋁ S), then similarly to the proofs of both (18) and (19), we have that c ∨ ⋁ S γ ≺ t ∨ ⋁ S γ(31) and atoms ( c ∨ ⋁ S γ ) = atoms ( t ∨ ⋁ S γ ) .(32)

If a _γ  ∈ atoms (c ∨ ⋁ S), then c ∨ ⋁ S = c ∨ ⋁ S _γ . Moreover, we have that a _γ  ∈ atoms (t ∨ ⋁ S) by formulas (18) and (19), and which implies that t ∨ ⋁ S = t ∨ ⋁ S _γ . Therefore, c ∨ ⋁ S γ ≺ t ∨ ⋁ S γ(33) and atoms ( c ∨ ⋁ S γ ) = atoms ( t ∨ ⋁ S γ )(34) by (29) and (30).

It is clear that (A, ⊆) is a wellordering. Thus, by Theorem 3.6 of [12] and formulas (26), (27), (31), (32), (33) and (44), we finally obtain t ∨ ⋁ S γ ≻ c ∨ ⋁ S γ and atoms ( t ∨ ⋁ S γ ) = atoms ( c ∨ ⋁ S γ ) for any γ ≤ α. Using formula (25), we know that atoms ( t ∨ ⋁ atoms ( g ) - atoms ( c ) ) = atoms ( c ∨ ⋁ atoms ( g ) - atoms ( c ) ) .

It is clear that c ∨ ⋁ atoms (g) - atoms (c) = g ∨ c, and this means that t ∨ ⋁ atoms (g) - atoms (c) = g ∨ t by (15). Hence, atoms (g ∨ t) = atoms (g ∨ c).

Therefore, in the case of g≰c, we have proven atoms (g ∨ t) = atoms (g ∨ c).

Cases (1) and (2) mean that atoms (c ∨ g) = atoms (t ∨ g) for any g ∈ G (L) with g ≤ z. Finally, from Definition 2.1 we know that the lattice L is WAGC. □

From the proof of Theorem 2.2, we have the following result.

Corollary 2.1. Every complete strongly atomic distributive lattice with the greatest element 1 = ⋁ atoms (L) satisfies the following condition.

(•) Let x, y, c ∈ G (L) and t ∈ L. If c = ⋁ atoms (x) ∩ atoms (y) ≺ t ≤ x ∧ y then any element z ∈ G (L) with t ≤ z satisfies condition it (C^∗).

(C^∗) atoms (c ∨ g) = atoms (t ∨ g) for any g ∈ G (L) with g ≤ z.

Theorem 2.3. If L is a strongly atomic semimodular WAGC lattice with the greatest element 1 = ⋁ atoms (L), then G (L) is a cover-preserving sublattice of L.

Proof. Suppose a, b ∈ G (L) satisfying a ≻ b in G (L). Then a = ⋁ atoms (a) and b = ⋁ atoms (b) by the definition of G (L) in Definition 2.1. Next, we shall show that G (L) is a cover-preserving sublattice of L, i.e., we shall prove that a ≻ b in L. There are two cases:

Case 1. If b = 0 then a ∈ atoms (L), thus a ≻ b in L.

Case 2. Suppose that b ≠ 0. We note that atoms (a) ⊋ atoms (b) since a ≻ b in G (L). Set x ∈ atoms (a) - atoms (b). Then we have that x ∨ b = ⋁ ( atoms ( b ) ∪ { x } ) > b ,(35) this means that x ∨ b ∈ G ( L ) .(36)

It is easy to see that x ∨ b ≤ a. Thus, by formulas (35) and (36) we have that a = x ∨ b since a ≻ b in G (L). Finally, we know that x ∧ b = 0 ≺ x in L, and which implies that x ∨ b ≻ b in L since L is semimodular. Hence, a ≻ b in L since a = x ∨ b.

Therefore, G (L) is a cover-preserving sublattice of L. □

Theorem 2.4. Every weakly locally, strongly atomic algebraic lattice L is semimodular if and only if it is a cover-preserving weakly locally semimodular lattice.

Proof. Suppose that L is cover-preserving weakly locally semimodular, and let a ∈ L and p₁, p₂ ∈ L with p₁ ≻ a, p₂ ≻ a and p₁ ≠ p₂. Note that a, p₁, p₂ and p₁ ∨ p₂ belong to the sublattice G ([a, u _a ]) by Definition 2.1, and G ([a, u _a ]) is semimodular since L is weakly locally semimodular. Hence, we easily check that p₁ ∨ p₂ covers p₁ and p₂ in G ([a, u _a ]). Again, G ([a, u _a ]) is a cover-preserving sublattice of L since L is cover-preserving weakly locally semimodular. Thus p₁ ∨ p₂ ≻ p₁ and p₁ ∨ p₂ ≻ p₂ in L. Therefore, L is semimodular by Theorem 3.7 of [3].

Conversely, suppose that the weakly locally, strongly atomic algebraic lattice L is semimodular. Then the sublattice [a, u _a ] for any a ∈ L is strongly atomic semimodular WAGC. By Theorem 2.3, we know that G ([a, u _a ]) for any a ∈ L is a cover-preserving sublattice of [a, u _a ], i.e., G ([a, u _a ]) is a cover-preserving sublattice of L.

Next, we prove that L is weakly locally semimodular. If L is not weakly locally semimodular, then there exists an element a ∈ L such that G ([a, u _a ]) is not semimodular. Thus we have two elements t, r ∈ G ([a, u _a ]) such that r ≻ r ∧ t but r ∨ t ⊁ t in G ([a, u _a ]). Hence r ≻ r ∧ t but r ∨ t ⊁ t in L since G ([a, u _a ]) is a cover-preserving sublattice of L. Therefore, the lattice L is not semimodular, contrary to the fact that L is semimodular. □

Theorem 2.5. Let L be a weakly locally, strongly atomic algebraic lattice which satisfies the following condition (Cr^∗). Then L is semimodular if and only if it is a cover-preserving weakly locally modular lattice.

it (Cr^∗) For all x, y ∈ L, if the sublattice [x, x ∨ y] has exactly one atom, then the interval [x ∧ y, y] has exactly one atom.

Note that Crawley [1961] denoted this condition by (ρ) (see [2]); Crawley and Dilworth [1973] denoted it by (∗) (see [3], p.53); M. Stern [1999] denoted it by (Cr^∗) (see [13], p. 174).

Proof. The sufficiency is clear, and we just prove the necessity. By Theorem 2.4, we know that if L is a weakly locally, strongly atomic algebraic semimodular lattice then L is a cover-preserving weakly locally semimodular lattice. Thus G ([a, u _a ]) is both a semimodular sublattice and a cover-preserving sublattice of L for any a ∈ L by Definition 2.2. Next, we only need to show that G ([a, u _a ]) is lower semimodular by Theorem 3.6 of [3], i.e., we shall show that G ([a, u _a ]) satisfies

x ≺ x ∨ y implies x ∧ y ≺ y for all x , y ∈ G ( [ a , u a ] ) .(37)

Assume that x ≺ x ∨ y in G ([a, u _a ]). We note that x, y, x ∧ y, x ∨ y ∈ G ([a, u _a ]), and x ≺ x ∨ y in L since G ([a, u _a ]) is a cover-preserving sublattice of L for any a ∈ L. By the condition (Cr^∗), the sublattice [x ∧ y, y] has exactly one atom, say z. We shall prove that z = y. On the contrary, suppose thatz < y.

We claim that there exists an element w ∈ G ([a, u _a ]) such that x ∧ y ≺ w ≤ y in G ([a, u _a ]). It is clear that x ∧ y < y since x ≺ x ∨ y. Thus there exists q ≻ a such that q ≤ y and q≰x ∧ y since both x ∧ y and y in G ([a, u _a ]). Hence, q ∧ (x ∧ y) = a ≺ q, and which means that q ∨ (x ∧ y) ≻ x ∧ y since G ([a, u _a ]) is a semimodular lattice. Let q ∨ (x ∧ y) = w. Clearly w ≤ y since q ≤ y and x ∧ y ≤ y. By Definition 2.1, we know that w ∈ G ([a, u _a ]) since q, x ∧ y ∈ G ([a, u _a ]). That is, w ∈ G ([a, u _a ]) and x ∧ y ≺ w ≤ y in G ([a, u _a ]).

Thus x ∧ y ≺ w ≤ y in L since G ([a, u _a ]) is a cover-preserving sublattice of L for any a ∈ L. However, the sublattice [x ∧ y, y] has exactly one atom z. Thus z = w ∈ G ([a, u _a ]). Since G ([a, u _a ]) is an atomic generated lattice and z < y in G ([a, u _a ]), we conclude that there is p ∈ G ([a, u _a ]) such that a ≺ p ≤ y and p≰z. Hence, a = p ∧ z ≥ p ∧ x ∧ y ≥ a, and which implies that p ∧ x ∧ y = a ≺ p. By semimodularity of G ([a, u _a ]) we deduce that x ∧ y ≺ p ∨ (x ∧ y) ≤ y in G ([a, u _a ]). Then x ∧ y ≺ p ∨ (x ∧ y) ≤ y in L since G ([a, u _a ]) is a cover-preserving sublattice of L for any a ∈ L. Hence z = p ∨ (x ∧ y) since z is the exactly atom of [x ∧ y, y], and this contradicts the fact that p≰z.

Consequently z = y, i.e., x ∧ y ≺ y. Therefore, the Equation (37) holds in G ([a, u _a ]) for any a ∈ L, and, in consequence, L is weakly locally modular. □

Inasmuch as we possess Theorem 2.5, we shall obtain the following corollary clearly.

Corollary 2.2. Let L be an atomic generated semimodular lattice. If L satisfies condition it (Cr^∗) then it is modular.

Note that the condition (Cr^∗) in Theorem 2.5 (resp. Corollary 2.2) can not be removed in general. It is clear that Fig. 2-5 is an atomic generated semimodular lattice in which the condition (Cr^∗) is not satisfied since [x, x ∨ y] has one atom but [x ∧ y, y] has two atoms p and q. Again, Fig. 2-5 is not a lower semimodular since x ∨ y ≻ x but y ⊁ x ∧ y. Hence Fig. 2-5 is not a modular lattice.

OPEN IN VIEWER

Fig.2-5

A non-modular atomic generated semimodular lattice L.

This paper presented two necessary and sufficient conditions that a strongly atomic algebraic lattice L is semimodular by using its WAGC sublattices. These necessary and sufficient conditions may be applying for describing the unique irredundant decompositions or replaceable irredundant decompositions of a semimodular lattice. For instance, from Theorems 7.5 of [3] and 2.5, we have that if a weakly locally, strongly atomic algebraic lattice L has replaceable irredundant decompositions, then L is semimodular if and only if L is a cover-preserving weakly locally modular lattice. From Theorem 7.5 of [3] and Corollary 2.2, every atomic generated semimodular lattice L has replaceable irredundant decompositions if and only if L is a modular lattice. Therefore, using the weakly local properties of a lattice to further investigate the unique irredundant decompositions or replaceable irredundant decompositions of a lattice is an interesting problem.