Page 220 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
A lattice T in a ring R with unity is centrally closed if the Boolean lattice of all central
idempotents lies in T. The previous result can be generalized as follows.
Corollary 6.1.13. Given a semisimple, Artinian ring R and a centrally closed lattice T in
R, the center Z(S) of the right extension S of T in R is the Boolean lattice of all complemented
elements in T.
Proof. This follows from the Wedderburn structure theorem for semisimple, Artinian ring and
the previous theorem. £
We conclude this section with a further consequence of the above results. Recall that a
skew chain is any skew lattice whose D-classes are totally ordered.
Theorem 6.1.14. Given R = Fn×n:
i) Every maximal right zero semigroup in R is a maximal rectangular band in R.
ii). Every maximal right-handed skew chain in R is a maximal skew lattice in R.
Proof. Consider a maximal right zero semigroup S given by all matrices of block form
⎡I j× j X⎤ . If S is not a maximal rectangular semigroup in R, then some β = ⎡I j× j 0⎤ exists
⎣⎢ 0 0 ⎥⎦ ⎣⎢ B 0⎥⎦
in E(R) \ S such that β together with S generates a properly larger rectangular band. But ⎡I X⎤
⎡ I 0⎤ is idempotent only if XB = 0 for all possible X. But this is not ⎣⎢0 0 ⎦⎥
⎢⎣B 0⎥⎦ = ⎡I + XB 0⎤ possible
⎢⎣ 0 0⎦⎥
since β ∈ S and thus B ≠ 0. Hence (i) must follow.
Suppose next that S is a maximal chain of right-rectangular skew lattices in R. Clearly S
is a quadratic skew lattice. Suppose that S lies in a larger skew lattice Sʹ. The inclusion S ⊆ Sʹ
cannot increase any D-class already in S. Hence a new D-class B exists in Sʹ that is
incomparable to some D-class A in S. If T is a lattice section of Sʹ, then Theorem 6.1.7
guarantees that the right extension of T will not include the full R-set A. (Put succinctly,
increasing the lattice shape of S decreases some of its D-class sizes.). Hence it is impossible to
join new D-class to S. £
Query: Given a lattice T in a matrix ring R of maximal height, is its right extension S a
maximal (right-handed) skew lattice in R? Conversely, are all maximal (right handed) skew
lattices in R of maximal height = n? Settling these questions even for matrix rings would be of
interest.
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A lattice T in a ring R with unity is centrally closed if the Boolean lattice of all central
idempotents lies in T. The previous result can be generalized as follows.
Corollary 6.1.13. Given a semisimple, Artinian ring R and a centrally closed lattice T in
R, the center Z(S) of the right extension S of T in R is the Boolean lattice of all complemented
elements in T.
Proof. This follows from the Wedderburn structure theorem for semisimple, Artinian ring and
the previous theorem. £
We conclude this section with a further consequence of the above results. Recall that a
skew chain is any skew lattice whose D-classes are totally ordered.
Theorem 6.1.14. Given R = Fn×n:
i) Every maximal right zero semigroup in R is a maximal rectangular band in R.
ii). Every maximal right-handed skew chain in R is a maximal skew lattice in R.
Proof. Consider a maximal right zero semigroup S given by all matrices of block form
⎡I j× j X⎤ . If S is not a maximal rectangular semigroup in R, then some β = ⎡I j× j 0⎤ exists
⎣⎢ 0 0 ⎥⎦ ⎣⎢ B 0⎥⎦
in E(R) \ S such that β together with S generates a properly larger rectangular band. But ⎡I X⎤
⎡ I 0⎤ is idempotent only if XB = 0 for all possible X. But this is not ⎣⎢0 0 ⎦⎥
⎢⎣B 0⎥⎦ = ⎡I + XB 0⎤ possible
⎢⎣ 0 0⎦⎥
since β ∈ S and thus B ≠ 0. Hence (i) must follow.
Suppose next that S is a maximal chain of right-rectangular skew lattices in R. Clearly S
is a quadratic skew lattice. Suppose that S lies in a larger skew lattice Sʹ. The inclusion S ⊆ Sʹ
cannot increase any D-class already in S. Hence a new D-class B exists in Sʹ that is
incomparable to some D-class A in S. If T is a lattice section of Sʹ, then Theorem 6.1.7
guarantees that the right extension of T will not include the full R-set A. (Put succinctly,
increasing the lattice shape of S decreases some of its D-class sizes.). Hence it is impossible to
join new D-class to S. £
Query: Given a lattice T in a matrix ring R of maximal height, is its right extension S a
maximal (right-handed) skew lattice in R? Conversely, are all maximal (right handed) skew
lattices in R of maximal height = n? Settling these questions even for matrix rings would be of
interest.
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