Page 91 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 91
III: Quasilattices, Paralattices and their Congruences
3.1 Congruences on quasilattices
Recall that a quasilattice is a noncommutative lattice (Q; ∨, ∧) for which the natural
quasi-orders (∨)≺ and (∧)≻ dualize each other: x (∨)≺ y iff x (∧)≻ y, that is, y∨x∨y = y iff x∧y∧x = x.
Corollary 1.3.5 asserts that a noncommutative lattice is a quasilattice if and only if D(∨) = D(∧)
with the common relation D being a congruence, in which case Q/D is the maximal lattice image
of Q and the D-classes are the maximal rectangular subalgebras of Q. Thus quasilattices are
precisely the class of those noncommutative lattices that follow the Clifford-McClean Theorem.
Indeed we might say that quasilattices are the most natural noncommutative generalizations of
lattices in that they have the rough shape of a lattice. They also have some striking properties that
have no precursors for regular bands.
Theorem 3.1.1. For any quasilattice (Q; ∨, ∧), the following hold:
i) For all congruences θ on Q, D o θ = θ o D = θ ∨ D.
ii) Given a family of congruences {θi} on Q:
D ∨ infi(θi) = infi(D ∨ θi) and D ∧ supi(θi) = supi(D ∧ θi).
Proof. Let x, y ∈Q be such that x Doθ y. Hence u ∈ Q exists such that x D u θ y. Set
w = (x∨y∨x) ∧ y ∧ (x∨y∨x). Then x = (x∨u∨x)∧u∧( x∨u∨x) θ w D y so that D o θ ⊆ θ o D. But
D = D–1 and θ = θ–1, so that θ o D = θ–1 o D –1 = (D o θ)–1 ⊆ (θ o D) –1 = D–1oθ–1 = D o θ,
and D o θ = θ o D follows with D o θ being precisely θ ∨ D.
Thanks to (i), we next need to show D o infi(θi) = infi(D o θi). That we have
D o infi(θi) ⊆ infi(D o θi) is clear. So let x infi(D o θi) y in Q. Then for each i, some ui in Q exists
such that x D ui θi y. Set w = (y∨x∨y)∧x∧(y∨x∨y). Then
y θi ui = (ui∧x∧ui)∨x∨(ui∧x∧ui) θ w D x
or just, x D w θi y. Since w works for all i, x D w (infiθi) y and infi(D o θi) ⊆ D o infi(θi) is seen.
Finally, clearly supi(D ∧ θi) ⊆ D ∧ supi(θi). So let x (D ∧ supi(θ)) y for some x, y ∈ Q.
Thus x D y and u1,… ,un exist such that x = u0 θ1 u1 θ2 u2 … un–1 θn un = y. For j ≤ n set
wj = (uj∧x∧uj)∨x∨(uj∧x∧uj). Then x D wj. In particular x = w0, y = wn, and wj –1 θj wj for j ≤ n.
Thus
x D∩θ1 w1 D∩θ2 w2 … wn–1 D∩θn y
so that x supi(D∩θi) y and supi(D∧θi) = D ∧ supi(θi) follows. £
As a consequence we have:
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3.1 Congruences on quasilattices
Recall that a quasilattice is a noncommutative lattice (Q; ∨, ∧) for which the natural
quasi-orders (∨)≺ and (∧)≻ dualize each other: x (∨)≺ y iff x (∧)≻ y, that is, y∨x∨y = y iff x∧y∧x = x.
Corollary 1.3.5 asserts that a noncommutative lattice is a quasilattice if and only if D(∨) = D(∧)
with the common relation D being a congruence, in which case Q/D is the maximal lattice image
of Q and the D-classes are the maximal rectangular subalgebras of Q. Thus quasilattices are
precisely the class of those noncommutative lattices that follow the Clifford-McClean Theorem.
Indeed we might say that quasilattices are the most natural noncommutative generalizations of
lattices in that they have the rough shape of a lattice. They also have some striking properties that
have no precursors for regular bands.
Theorem 3.1.1. For any quasilattice (Q; ∨, ∧), the following hold:
i) For all congruences θ on Q, D o θ = θ o D = θ ∨ D.
ii) Given a family of congruences {θi} on Q:
D ∨ infi(θi) = infi(D ∨ θi) and D ∧ supi(θi) = supi(D ∧ θi).
Proof. Let x, y ∈Q be such that x Doθ y. Hence u ∈ Q exists such that x D u θ y. Set
w = (x∨y∨x) ∧ y ∧ (x∨y∨x). Then x = (x∨u∨x)∧u∧( x∨u∨x) θ w D y so that D o θ ⊆ θ o D. But
D = D–1 and θ = θ–1, so that θ o D = θ–1 o D –1 = (D o θ)–1 ⊆ (θ o D) –1 = D–1oθ–1 = D o θ,
and D o θ = θ o D follows with D o θ being precisely θ ∨ D.
Thanks to (i), we next need to show D o infi(θi) = infi(D o θi). That we have
D o infi(θi) ⊆ infi(D o θi) is clear. So let x infi(D o θi) y in Q. Then for each i, some ui in Q exists
such that x D ui θi y. Set w = (y∨x∨y)∧x∧(y∨x∨y). Then
y θi ui = (ui∧x∧ui)∨x∨(ui∧x∧ui) θ w D x
or just, x D w θi y. Since w works for all i, x D w (infiθi) y and infi(D o θi) ⊆ D o infi(θi) is seen.
Finally, clearly supi(D ∧ θi) ⊆ D ∧ supi(θi). So let x (D ∧ supi(θ)) y for some x, y ∈ Q.
Thus x D y and u1,… ,un exist such that x = u0 θ1 u1 θ2 u2 … un–1 θn un = y. For j ≤ n set
wj = (uj∧x∧uj)∨x∨(uj∧x∧uj). Then x D wj. In particular x = w0, y = wn, and wj –1 θj wj for j ≤ n.
Thus
x D∩θ1 w1 D∩θ2 w2 … wn–1 D∩θn y
so that x supi(D∩θi) y and supi(D∧θi) = D ∧ supi(θi) follows. £
As a consequence we have:
89
