Page 223 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 223
VI: Skew Lattices in Rings
Theorem 6.2.3. Join classes and meet classes are given by commuting joins and meets.
Thus given D-classes A and B in a s-band S, their join D -class J and meet D -class M are
J = {asb⎮a ∈ A, b ∈ B & asb = bsa} and M = {ab⎮a ∈ A, b ∈ B & ab = ba}.
Moreover, for every a ∈ A there exists b ∈ B such that asb = bsa in J and ab = ba in M.
Proof. Given v ∈ J, there exist a ∈ A and b ∈ B such that v ≥ a, b. (Set a = vaʹv and b = vbʹv for
any aʹ ∈ A and bʹ ∈ B.) For such a and b we have a∇b ∈ J and
v = v(asb)v = v(a + b + ba – aba – bab)v = a + b + ba – aba – bab = asb.
Similarly, bsa equals v also and the assertion about J is seen. The case for M is similar. For the
final assertion, pick a in A and let v ∈ J be such that v ≥ a. That b ∈ B exists such that
asb = v = bsa is now clear. The rest follows from symmetry. £
Corollary 6.2.4. Given a s-band S and e ∈ S, the following are equivalent:
i) De = {e}.
ii) For all x ∈ S, esx = xse and ex = xe.
Proof. Clearly (ii) implies (i); and (ii) implies (i) due to the final assertion of Theorem 6.2.3. £
Corollary 6.2.5. A set of commuting elements in a s-band S generates a sublattice. £
We next turn to the question when s is associative, giving various criteria. When it is
associative on a particular s-band, the latter is called a cubic skew lattice in the ring. Clearly
quadratic skew lattices in a ring (studied in the previous section) are trivially cubic.
The associativity of s: the role of the commutator [x, y]
Recall that the commutator of elements x, y in a ring R is [x, y] = xy – yx. Clearly x and y
commute if and only if [x, y] = 0. For any pair of idempotents e and f in a band S in a ring R
esf – e○f = ef + fe – efe – fef = [e, f]2.
Thus [e, f]2 = 0 on a s-band S if and only if s = ○ as binary operations on S. In this section we
show that a s-band S is associative if and only if for all e, f ∈ S, [e, f]2 lies in the center of the
subring of R generated from S. That is, s is associative if and only if g[e, f]2 = [e, f]2g for all e,
f, g ∈ S. We begin with a pair of somewhat technical lemmas.
Lemma 6.2.6. (asb)c(asb) = – (abca – aca – bca – bcb + bcab).
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Theorem 6.2.3. Join classes and meet classes are given by commuting joins and meets.
Thus given D-classes A and B in a s-band S, their join D -class J and meet D -class M are
J = {asb⎮a ∈ A, b ∈ B & asb = bsa} and M = {ab⎮a ∈ A, b ∈ B & ab = ba}.
Moreover, for every a ∈ A there exists b ∈ B such that asb = bsa in J and ab = ba in M.
Proof. Given v ∈ J, there exist a ∈ A and b ∈ B such that v ≥ a, b. (Set a = vaʹv and b = vbʹv for
any aʹ ∈ A and bʹ ∈ B.) For such a and b we have a∇b ∈ J and
v = v(asb)v = v(a + b + ba – aba – bab)v = a + b + ba – aba – bab = asb.
Similarly, bsa equals v also and the assertion about J is seen. The case for M is similar. For the
final assertion, pick a in A and let v ∈ J be such that v ≥ a. That b ∈ B exists such that
asb = v = bsa is now clear. The rest follows from symmetry. £
Corollary 6.2.4. Given a s-band S and e ∈ S, the following are equivalent:
i) De = {e}.
ii) For all x ∈ S, esx = xse and ex = xe.
Proof. Clearly (ii) implies (i); and (ii) implies (i) due to the final assertion of Theorem 6.2.3. £
Corollary 6.2.5. A set of commuting elements in a s-band S generates a sublattice. £
We next turn to the question when s is associative, giving various criteria. When it is
associative on a particular s-band, the latter is called a cubic skew lattice in the ring. Clearly
quadratic skew lattices in a ring (studied in the previous section) are trivially cubic.
The associativity of s: the role of the commutator [x, y]
Recall that the commutator of elements x, y in a ring R is [x, y] = xy – yx. Clearly x and y
commute if and only if [x, y] = 0. For any pair of idempotents e and f in a band S in a ring R
esf – e○f = ef + fe – efe – fef = [e, f]2.
Thus [e, f]2 = 0 on a s-band S if and only if s = ○ as binary operations on S. In this section we
show that a s-band S is associative if and only if for all e, f ∈ S, [e, f]2 lies in the center of the
subring of R generated from S. That is, s is associative if and only if g[e, f]2 = [e, f]2g for all e,
f, g ∈ S. We begin with a pair of somewhat technical lemmas.
Lemma 6.2.6. (asb)c(asb) = – (abca – aca – bca – bcb + bcab).
221
