Page 105 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 105
III: Quasilattices, Paralattices and their Congruences
Moreover, bijections fα: Dα → Dαʹ between all corresponding D-classes of Q and Qʹ are induced
upon restricting f to the various D-classes of S. Conversely, given quasilattices (Q, ∨, ∧) and
(Qʹ, ∨ʹ, ∧ʹ), suppose that f*: Q/D → Qʹ/D is an isomorphism and that bijections fα: Dα → Dαʹ exist
between all corresponding pairs of D-classes (relative to f*). Then an isotopy f: Q → Qʹ is given
by f = ∪ fα. Any isotopy of (Q, ∨, ∧) with (Qʹ, ∨ʹ, ∧ʹ) thus arises in this manner. £
The existence of an isotopy between quasilattices thus depends on having isomorphic
maximal lattice images and corresponding D-classes of the same size. For refined quasilattices
more constraints occur.
Proposition 3.4.11. An isotopy f: (S, ∨, ∧) → (Sʹ, ∨ʹ, ∧ʹ) of refined quasilattices preserves
both commuting joins and meets. Thus a∨b = b∨a in S implies f(a∨b) = f(a) ∨ʹ f(b) = f(b) ∨ʹ f(a) in
Sʹ. Dually, a∧b = b∧a in S implies f(a∧b) = f(a) ∧ʹ f(b) = f(b) ∧ʹ f(a) in Sʹ.
Proof. Suppose that a∨b = b∨a in S. If c denotes this common join, then c is the unique element
in the join-class of the D-classes of a and b such that both c ≥ a and c ≥ b. Being an isotropy of
paralattices, we have f(c) ≥ both f(a) and f(b) in S ʹ. Being an isotropy of quasilattices, we have
f(c) lying in the join class of the D-classes of f(a) and f(b) in Sʹ. By uniqueness,
f(a) ∨ʹ f(b) = f(c) = f(b) ∨ʹ f(a). Similarly, f preserves commuting meets. £
Michael Kinyon observed (in a personal comminication) that every refined quasilattice is
isotopic to a skew lattice in a fairly simple way. To see this first recall a fact about regular bands.
Lemma 3.4.12. If (S, •) is a band and an operation •L is defined on S by e•Lf = efe, then
(S, •L) is a band if and only if (S, •) is a regular band, in which case (S, •L) is a left regular band.
Dually, defining •R on S by e•Rf = fef, (S, •R) is a band if and only if (S, •) is a regular band, in
which case the band (S, •R) is right regular. Moreover, e ≤ f in (S, •) iff e ≤ f in (S, •L), and e ≺ f
in (S, •) iff e ≺ f in (S, •L). Similar remarks relate both ≤ and ≺ for (S, •) and (S, •R).
Proof. (Here x•y is denoted by xy.) Clearly •L is idempotent. Is it associative? Note that
a•L(b•Lc) = a•L(bcb) = abcba while (a•Lb)•Lc = (a•Lb)c(a•Lb) = abacaba. Thus •L is associative
iff
abcba = abacaba
holds on S. Clearly this happens if (S, •) is regular. Conversely, if (S, •L) is associative, then
expanding a•Lb•Lc both ways gives abcba = abacaba for all a, b, c. Replace a by ba to get
babcbba = babbacbabba, which simplifies to
The mirror argument gives (1) babcba = bacba
(2) abcbab = abcab.
103
Moreover, bijections fα: Dα → Dαʹ between all corresponding D-classes of Q and Qʹ are induced
upon restricting f to the various D-classes of S. Conversely, given quasilattices (Q, ∨, ∧) and
(Qʹ, ∨ʹ, ∧ʹ), suppose that f*: Q/D → Qʹ/D is an isomorphism and that bijections fα: Dα → Dαʹ exist
between all corresponding pairs of D-classes (relative to f*). Then an isotopy f: Q → Qʹ is given
by f = ∪ fα. Any isotopy of (Q, ∨, ∧) with (Qʹ, ∨ʹ, ∧ʹ) thus arises in this manner. £
The existence of an isotopy between quasilattices thus depends on having isomorphic
maximal lattice images and corresponding D-classes of the same size. For refined quasilattices
more constraints occur.
Proposition 3.4.11. An isotopy f: (S, ∨, ∧) → (Sʹ, ∨ʹ, ∧ʹ) of refined quasilattices preserves
both commuting joins and meets. Thus a∨b = b∨a in S implies f(a∨b) = f(a) ∨ʹ f(b) = f(b) ∨ʹ f(a) in
Sʹ. Dually, a∧b = b∧a in S implies f(a∧b) = f(a) ∧ʹ f(b) = f(b) ∧ʹ f(a) in Sʹ.
Proof. Suppose that a∨b = b∨a in S. If c denotes this common join, then c is the unique element
in the join-class of the D-classes of a and b such that both c ≥ a and c ≥ b. Being an isotropy of
paralattices, we have f(c) ≥ both f(a) and f(b) in S ʹ. Being an isotropy of quasilattices, we have
f(c) lying in the join class of the D-classes of f(a) and f(b) in Sʹ. By uniqueness,
f(a) ∨ʹ f(b) = f(c) = f(b) ∨ʹ f(a). Similarly, f preserves commuting meets. £
Michael Kinyon observed (in a personal comminication) that every refined quasilattice is
isotopic to a skew lattice in a fairly simple way. To see this first recall a fact about regular bands.
Lemma 3.4.12. If (S, •) is a band and an operation •L is defined on S by e•Lf = efe, then
(S, •L) is a band if and only if (S, •) is a regular band, in which case (S, •L) is a left regular band.
Dually, defining •R on S by e•Rf = fef, (S, •R) is a band if and only if (S, •) is a regular band, in
which case the band (S, •R) is right regular. Moreover, e ≤ f in (S, •) iff e ≤ f in (S, •L), and e ≺ f
in (S, •) iff e ≺ f in (S, •L). Similar remarks relate both ≤ and ≺ for (S, •) and (S, •R).
Proof. (Here x•y is denoted by xy.) Clearly •L is idempotent. Is it associative? Note that
a•L(b•Lc) = a•L(bcb) = abcba while (a•Lb)•Lc = (a•Lb)c(a•Lb) = abacaba. Thus •L is associative
iff
abcba = abacaba
holds on S. Clearly this happens if (S, •) is regular. Conversely, if (S, •L) is associative, then
expanding a•Lb•Lc both ways gives abcba = abacaba for all a, b, c. Replace a by ba to get
babcbba = babbacbabba, which simplifies to
The mirror argument gives (1) babcba = bacba
(2) abcbab = abcab.
103
