Page 221 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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VI: Skew Lattices in Rings
6.2 ∇-bands and cubic skew lattices
Given a ring R and e, f, in E(R) recall that e∇f = (e ○ f)2 = (e + f – ef)2. The latter
expands to e + f + fe – ef – efe – fef + efef in general, but reduces to e + f + fe – efe – fef when ef is
idempotent. Recall also that ∇ extends ○ in that:
i) Every skew lattice (S; ○, •) in a ring is also a skew lattice under ∇ and • since in
this case (e ○ f)2 = e ○ f so that e∇f reduces to e ○ f.
ii) Whenever e, f, ef, fe ∈E(R), then so are efe, fef and e∇f by Theorem 2.7.5.
iii) Situations occur where e∇f is idempotent, but not e ○ f; but ∇ need not be
associative, even if idempotent closure occurs. (See Examples 2.3.1 and 2.3.2.)
iv). Due to (i), maximal right [left] regular bands in any ring form skew lattices under
∇ and •. Every right [left] regular band in a ring generates a skew lattice under ∇
and •.
v). Every maximal normal band in a ring R is a normal skew lattice under ∇ and •;
indeed it forms a skew Boolean algebra. Thus a normal band in R generates a
strongly distributive skew lattice under ∇ and •. (Theorem 2.3.6)
vi) Maximal regular bands in rings, however, need not be closed under ∇, much less
be skew lattices under ∇. (See Example 2.3.5.)
Recall that every band is naturally partially ordered by e ≤ f if ef = e = fe which refines
the natural preorder given by e ≺ f if efe = e. The equivalence induced from ≺ is the Green’s
relation D. Turning to s-bands proper, a band congruence θ on a s-band is a s-congruence if
θ is also a congruence under s. The assertions in the following lemma, coming from a 2004
paper of Cvetko-Vah, are easily verified.
Lemma 6.2.1. Given a s-band S in a ring, for all a, b ∈ S:
i) Given b ≺ a in S, a∇b = a + ba – aba and b∇a = a + ab – aba.
ii) a∇b∇a is unambiguous: (a∇b)∇a = a∇(b∇a) = a + b – bab. £
Even when s is not associative, s-bands are very much like skew lattices in rings.
Theorem 6.2.2. For any s- band S in a ring the following hold.
i) As a band, S is regular. That is, abaca = abca holds on S.
ii) D is a s-congruence and S/D is a lattice with Dx ∨ Dy = Dxsy.
iii) a (asb) = a = (bsa)a and as(ab) = a = (ba)sa.
iv) ab = ba iff asb = bsa.
v) a (bsc) a = aba s aca and as(bc)sa = (asbsa)(ascsa).
vi) a∇c∇a = b∇c∇b and aca = bcb implies a = b.
(S; s, •) is thus a distributive, symmetric, cancellative skew lattice when s is also associative.
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6.2 ∇-bands and cubic skew lattices
Given a ring R and e, f, in E(R) recall that e∇f = (e ○ f)2 = (e + f – ef)2. The latter
expands to e + f + fe – ef – efe – fef + efef in general, but reduces to e + f + fe – efe – fef when ef is
idempotent. Recall also that ∇ extends ○ in that:
i) Every skew lattice (S; ○, •) in a ring is also a skew lattice under ∇ and • since in
this case (e ○ f)2 = e ○ f so that e∇f reduces to e ○ f.
ii) Whenever e, f, ef, fe ∈E(R), then so are efe, fef and e∇f by Theorem 2.7.5.
iii) Situations occur where e∇f is idempotent, but not e ○ f; but ∇ need not be
associative, even if idempotent closure occurs. (See Examples 2.3.1 and 2.3.2.)
iv). Due to (i), maximal right [left] regular bands in any ring form skew lattices under
∇ and •. Every right [left] regular band in a ring generates a skew lattice under ∇
and •.
v). Every maximal normal band in a ring R is a normal skew lattice under ∇ and •;
indeed it forms a skew Boolean algebra. Thus a normal band in R generates a
strongly distributive skew lattice under ∇ and •. (Theorem 2.3.6)
vi) Maximal regular bands in rings, however, need not be closed under ∇, much less
be skew lattices under ∇. (See Example 2.3.5.)
Recall that every band is naturally partially ordered by e ≤ f if ef = e = fe which refines
the natural preorder given by e ≺ f if efe = e. The equivalence induced from ≺ is the Green’s
relation D. Turning to s-bands proper, a band congruence θ on a s-band is a s-congruence if
θ is also a congruence under s. The assertions in the following lemma, coming from a 2004
paper of Cvetko-Vah, are easily verified.
Lemma 6.2.1. Given a s-band S in a ring, for all a, b ∈ S:
i) Given b ≺ a in S, a∇b = a + ba – aba and b∇a = a + ab – aba.
ii) a∇b∇a is unambiguous: (a∇b)∇a = a∇(b∇a) = a + b – bab. £
Even when s is not associative, s-bands are very much like skew lattices in rings.
Theorem 6.2.2. For any s- band S in a ring the following hold.
i) As a band, S is regular. That is, abaca = abca holds on S.
ii) D is a s-congruence and S/D is a lattice with Dx ∨ Dy = Dxsy.
iii) a (asb) = a = (bsa)a and as(ab) = a = (ba)sa.
iv) ab = ba iff asb = bsa.
v) a (bsc) a = aba s aca and as(bc)sa = (asbsa)(ascsa).
vi) a∇c∇a = b∇c∇b and aca = bcb implies a = b.
(S; s, •) is thus a distributive, symmetric, cancellative skew lattice when s is also associative.
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