Page 18 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 18
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Of particular interest is the next result. It’s proof may be obtained in any standard text on
lattice theory.
Theorem 1.1.5. Congruence lattices of lattices are distributive. £
A subset U of a poset (L; ≥) is directed upward if given any two elements x, y in U, a
third element z exists in U such that x, y ≤ z. The proof of the next result is also easily accessible.
Theorem 1.1.6. Given an algebraic lattice (L; ∨, ∧), a ∧ sup(U) = sup{a∧x ⎢x ∈ U}
holds if U is directed upward. This equality holds unconditionally when (L; ∨, ∧) is also
distributive.
Recall that two algebras A = (A; f1, f2, …, fr) and B = (B; g1, g2, …, gs) have the same
type if r = s and for all i ≤ r, both fi and gi have the same number of variables, that is, both are say
ni-ary operations. Recall also that a class V of algebras of the same type is a variety if it is closed
under direct products, subalgebras and homomorphic images. A classic result of Birkhoff is as
follows:
Theorem 1.1.7. Among algebras of the same type, each variety is determined by the set
of all identities satisfied by all algebras in that variety. That is, all varieties are equationally
determined in the class of all algebras of the same type. £
Let B = {Bi ⎢i ∈ I} be a set of algebras of the same type. An algebra A of the same type
is a subdirect product of the Bi if a monomorphism χ: A → ∏i∈iBi exists such that for each
projection πi: ∏i∈IBi → Bi, the composite πioχ: A → Bi goes onto Bi. A is subdirectly
irreducible if for any subdirect factorization χ: A → ∏i∈IBi one of the composites πi o χ: A → Bi
is an isomorphism. A second classic result of Birkhoff is as follows:
Theorem 1.1.8. In a given variety of algebras V, every algebra A in V is a subdirect
product of subdirectly irreducible algebras. £
We apply Theorem 1.1.8 to the variety of distributive lattices. But first recall that an
ideal in a lattice L is any subset I of L that is closed joins and given any x ∈ I and y ∈ L, x∧y ∈ I
also. Recall also that a filter (or dual-ideal) in a lattice L is any subset F of L that is closed under
meets and given any x ∈ F and y ∈ L, x∨y ∈ F also. Given any element x ∈ L, the principal ideal
x↓ = {y ∈ L⎜x ≥ y} is the smallest ideal of L containing x. Dually, the smallest filter of L
containing x is the principal filter x↑ = {y ∈ L⎜x ≤ y}.
Theorem 1.1.9. Given a distributive lattice (L; ∨, ∧) and an element a ∈ L,
χ: L → a↓ × a↑ defined by χ(x) = (x∧a, x∨a) is a subdirect decomposition of (L; ∨, ∧). Thus a
distributive lattice is subdirectly irreducible if and only if it is a copy of either C0 or C1.
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Of particular interest is the next result. It’s proof may be obtained in any standard text on
lattice theory.
Theorem 1.1.5. Congruence lattices of lattices are distributive. £
A subset U of a poset (L; ≥) is directed upward if given any two elements x, y in U, a
third element z exists in U such that x, y ≤ z. The proof of the next result is also easily accessible.
Theorem 1.1.6. Given an algebraic lattice (L; ∨, ∧), a ∧ sup(U) = sup{a∧x ⎢x ∈ U}
holds if U is directed upward. This equality holds unconditionally when (L; ∨, ∧) is also
distributive.
Recall that two algebras A = (A; f1, f2, …, fr) and B = (B; g1, g2, …, gs) have the same
type if r = s and for all i ≤ r, both fi and gi have the same number of variables, that is, both are say
ni-ary operations. Recall also that a class V of algebras of the same type is a variety if it is closed
under direct products, subalgebras and homomorphic images. A classic result of Birkhoff is as
follows:
Theorem 1.1.7. Among algebras of the same type, each variety is determined by the set
of all identities satisfied by all algebras in that variety. That is, all varieties are equationally
determined in the class of all algebras of the same type. £
Let B = {Bi ⎢i ∈ I} be a set of algebras of the same type. An algebra A of the same type
is a subdirect product of the Bi if a monomorphism χ: A → ∏i∈iBi exists such that for each
projection πi: ∏i∈IBi → Bi, the composite πioχ: A → Bi goes onto Bi. A is subdirectly
irreducible if for any subdirect factorization χ: A → ∏i∈IBi one of the composites πi o χ: A → Bi
is an isomorphism. A second classic result of Birkhoff is as follows:
Theorem 1.1.8. In a given variety of algebras V, every algebra A in V is a subdirect
product of subdirectly irreducible algebras. £
We apply Theorem 1.1.8 to the variety of distributive lattices. But first recall that an
ideal in a lattice L is any subset I of L that is closed joins and given any x ∈ I and y ∈ L, x∧y ∈ I
also. Recall also that a filter (or dual-ideal) in a lattice L is any subset F of L that is closed under
meets and given any x ∈ F and y ∈ L, x∨y ∈ F also. Given any element x ∈ L, the principal ideal
x↓ = {y ∈ L⎜x ≥ y} is the smallest ideal of L containing x. Dually, the smallest filter of L
containing x is the principal filter x↑ = {y ∈ L⎜x ≤ y}.
Theorem 1.1.9. Given a distributive lattice (L; ∨, ∧) and an element a ∈ L,
χ: L → a↓ × a↑ defined by χ(x) = (x∧a, x∨a) is a subdirect decomposition of (L; ∨, ∧). Thus a
distributive lattice is subdirectly irreducible if and only if it is a copy of either C0 or C1.
16
