Page 15 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 15
I: Preliminaries
monograph, Cell complexes, poset tolopogy and the representation theory of algebras arising in
algebraic combinatorics and discrete geometry by Stuart Margolis, Franco Saliola and Benjamin
Steinberg.
1.1 Lattices
Recall that a partially ordered set or poset is any pair (L; ≥) where L is a set and ≥ is a
partial ordering of L, that is, reflexive, anti-symmetric and transitive relation on L. Given x ≥ y in
L, we think of x as lying above y, or equally of y lying below x.
Given elements x, y in a poset (L; ≥), an element m ∈ L such that (1) x ≥ m and y ≥ m and
(2) m lies above all other elements lying jointly below x and y is called the meet of x and y and is
denoted by x∧y. Dually, an element j ∈ L such that (3) j ≥ x, j ≥ y and (4) j lies below all other
elements lying jointly above x and y is called the join of x and y and is denoted by x∨y.
x∨y
xy
x∧y
When they exist, x∧y and x∨y are unique with respect to the given x and y. If all pairs x, y ∈ S
have a meet and a join, then (L; ≥) is a lattice. In this case (L; ∨, ∧) satisfies the following
idempotent, commutative, associative and absorption identities:
L0. x∧x = x = x∨x.
L1. x∧y = y∧x and x∨y = y∨x.
L2. (x∧y)∧z = x∧(y∧z) and (x∨y)∨z = x∨(y∨z).
L3. x∧(x∨y) = x = x∨(x∧y).
Conversely, given (L; ∨, ∧) with binary operations ∧ and ∨ satisfying L0 – L3, a partial order ≥ is
defined on L by
x ≥ y if and only if x ∧ y = y, or equivalently, x ∨ y = y.
As a poset, (L; ≥) is a lattice whose meets and joins are precisely the given ∧ and ∨. Indeed the
process of passing from a lattice poset (L; ≥) to an algebra (L; ∨, ∧) and the reverse process of
passing from an algebra (L; ∨, ∧) satisfying L0-L3 to a lattice poset are reciprocal processes.
Thus lattices may be viewed from either a poset perspective or an algebraic perspective.
While not verifying all details, we offer the following remarks. To begin, in any lattice
poset, L0 and L1 are clear. L2 refers to the unique elements x∧y∧z lying maximally below x, y
and z and x∨y∨z lying minimally above x, y and z. Likewise, the absorption identities in L3 refer
to the fact that x∨y ≥ x ≥ x∧y in (L; ≥). Conversely, given an algebra (L; ∨, ∧) satisfying L0 – L3,
the derived relation ≥ is reflexive by L0 and anti-symmetric thanks to L1. L2 is instrumental in
13
monograph, Cell complexes, poset tolopogy and the representation theory of algebras arising in
algebraic combinatorics and discrete geometry by Stuart Margolis, Franco Saliola and Benjamin
Steinberg.
1.1 Lattices
Recall that a partially ordered set or poset is any pair (L; ≥) where L is a set and ≥ is a
partial ordering of L, that is, reflexive, anti-symmetric and transitive relation on L. Given x ≥ y in
L, we think of x as lying above y, or equally of y lying below x.
Given elements x, y in a poset (L; ≥), an element m ∈ L such that (1) x ≥ m and y ≥ m and
(2) m lies above all other elements lying jointly below x and y is called the meet of x and y and is
denoted by x∧y. Dually, an element j ∈ L such that (3) j ≥ x, j ≥ y and (4) j lies below all other
elements lying jointly above x and y is called the join of x and y and is denoted by x∨y.
x∨y
xy
x∧y
When they exist, x∧y and x∨y are unique with respect to the given x and y. If all pairs x, y ∈ S
have a meet and a join, then (L; ≥) is a lattice. In this case (L; ∨, ∧) satisfies the following
idempotent, commutative, associative and absorption identities:
L0. x∧x = x = x∨x.
L1. x∧y = y∧x and x∨y = y∨x.
L2. (x∧y)∧z = x∧(y∧z) and (x∨y)∨z = x∨(y∨z).
L3. x∧(x∨y) = x = x∨(x∧y).
Conversely, given (L; ∨, ∧) with binary operations ∧ and ∨ satisfying L0 – L3, a partial order ≥ is
defined on L by
x ≥ y if and only if x ∧ y = y, or equivalently, x ∨ y = y.
As a poset, (L; ≥) is a lattice whose meets and joins are precisely the given ∧ and ∨. Indeed the
process of passing from a lattice poset (L; ≥) to an algebra (L; ∨, ∧) and the reverse process of
passing from an algebra (L; ∨, ∧) satisfying L0-L3 to a lattice poset are reciprocal processes.
Thus lattices may be viewed from either a poset perspective or an algebraic perspective.
While not verifying all details, we offer the following remarks. To begin, in any lattice
poset, L0 and L1 are clear. L2 refers to the unique elements x∧y∧z lying maximally below x, y
and z and x∨y∨z lying minimally above x, y and z. Likewise, the absorption identities in L3 refer
to the fact that x∨y ≥ x ≥ x∧y in (L; ≥). Conversely, given an algebra (L; ∨, ∧) satisfying L0 – L3,
the derived relation ≥ is reflexive by L0 and anti-symmetric thanks to L1. L2 is instrumental in
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