Page 17 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 17
I: Preliminaries
Proof. Given x∧z = y∧z and x∨z = y∨z, then
x = x ∨ (x ∧ z) = x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) = (x ∨ y) ∧ (y ∨ z) = (x ∨ z) ∧ y ≤ y
and similarly, y ≤ x, so that x = y. Conversely, neither M3 nor N5 can be subalgebras of a
cancellative slew lattice. £
A lattice (L; ∨, ∧) is complete if every subset X of L has a supremum (an element u ≥ x
for all x in X, with u being the least such element in L) denoted by sup(X) and an infimum (an
element v ≤ x for all x in X, with v being the greatest such element in L) denoted by inf(X). In
particular, a complete lattice has a greatest element 1 and a least element 0. Conversely, a lattice
with both least and greatest elements 0 and 1 is complete if all subsets have suprema, or
equivalently, if all subsets have infima. Finally, in any complete lattice, we let 0 = sup(∅) and
1 = inf(∅).
Lattices and universal algebra
An algebra is any system, A = (A: f1, f2, …, fr), where A is a set and each fi is an ni-ary
operation on A. If B ⊆ A is such that for all i ≤ r, fi(b1, b2, …, bni ) ∈ B for all b1, …, bni in B,
then the system B = (B: f1ʹ, f2ʹ, …, frʹ) where fiʹ= fi ⎢ Bni is a subalgebra of A. (When confusion
occurs, subalgebras may be indicated by their underlying sets.) Under inclusion, ⊆, the
subalgebras of an algebra A form a complete lattice Sub(A) with greatest element A, least
element the smallest subalgebra containing ∅ and meets given by intersection. If none of the
operations are nullary, then the least subalgebra is the empty subalgebra, ∅. If there are no
operations, then Sub(A) is the lattice 2A.
Recall that a congruence on A = (A: f1, f2, …, fr) is an equivalence relation θ on A such
that given i ≤ r with a1θb1, a2θb2, …, ani θ bni in A, then
fi(a1, a2, …, ani ) θ fi(b1, b2, …, bni ).
Under inclusion, ⊆, the congruences on A form a complete lattice Con(A). Its greatest element is
the universal relation ∇ = A×A relating all elements in A. Its least element is the identity relation
Δ. Suprema and infima in Con(A) are calculated as in the lattice Equ(A) of all equivalences on
A. In particular, infima in Con(A) are given by intersection. £
Recall that an element c in a lattice (L; ∨, ∧) is compact if for any subset X of L, c ≤ supX
implies that c ≤ supY for some finite subset Y of X. (Every cover can be reduced to a finite cover.)
An algebraic lattice is a complete lattice for which every element is a supremum of compact
elements. The proof of the following result is easily accessible in the literature
Theorem 1.1.4. Given an algebra A = (A: f1, f2, …, fr), both Sub(A) and Con(A) are
algebraic lattices.
15
Proof. Given x∧z = y∧z and x∨z = y∨z, then
x = x ∨ (x ∧ z) = x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) = (x ∨ y) ∧ (y ∨ z) = (x ∨ z) ∧ y ≤ y
and similarly, y ≤ x, so that x = y. Conversely, neither M3 nor N5 can be subalgebras of a
cancellative slew lattice. £
A lattice (L; ∨, ∧) is complete if every subset X of L has a supremum (an element u ≥ x
for all x in X, with u being the least such element in L) denoted by sup(X) and an infimum (an
element v ≤ x for all x in X, with v being the greatest such element in L) denoted by inf(X). In
particular, a complete lattice has a greatest element 1 and a least element 0. Conversely, a lattice
with both least and greatest elements 0 and 1 is complete if all subsets have suprema, or
equivalently, if all subsets have infima. Finally, in any complete lattice, we let 0 = sup(∅) and
1 = inf(∅).
Lattices and universal algebra
An algebra is any system, A = (A: f1, f2, …, fr), where A is a set and each fi is an ni-ary
operation on A. If B ⊆ A is such that for all i ≤ r, fi(b1, b2, …, bni ) ∈ B for all b1, …, bni in B,
then the system B = (B: f1ʹ, f2ʹ, …, frʹ) where fiʹ= fi ⎢ Bni is a subalgebra of A. (When confusion
occurs, subalgebras may be indicated by their underlying sets.) Under inclusion, ⊆, the
subalgebras of an algebra A form a complete lattice Sub(A) with greatest element A, least
element the smallest subalgebra containing ∅ and meets given by intersection. If none of the
operations are nullary, then the least subalgebra is the empty subalgebra, ∅. If there are no
operations, then Sub(A) is the lattice 2A.
Recall that a congruence on A = (A: f1, f2, …, fr) is an equivalence relation θ on A such
that given i ≤ r with a1θb1, a2θb2, …, ani θ bni in A, then
fi(a1, a2, …, ani ) θ fi(b1, b2, …, bni ).
Under inclusion, ⊆, the congruences on A form a complete lattice Con(A). Its greatest element is
the universal relation ∇ = A×A relating all elements in A. Its least element is the identity relation
Δ. Suprema and infima in Con(A) are calculated as in the lattice Equ(A) of all equivalences on
A. In particular, infima in Con(A) are given by intersection. £
Recall that an element c in a lattice (L; ∨, ∧) is compact if for any subset X of L, c ≤ supX
implies that c ≤ supY for some finite subset Y of X. (Every cover can be reduced to a finite cover.)
An algebraic lattice is a complete lattice for which every element is a supremum of compact
elements. The proof of the following result is easily accessible in the literature
Theorem 1.1.4. Given an algebra A = (A: f1, f2, …, fr), both Sub(A) and Con(A) are
algebraic lattices.
15
