Page 218 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 218
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In what follows, without loss of generality we assume that T is a sublattice of Δ such that
both matrices 0 and I lie in T. With this in mind, for each e ∈ Δ,
the T-cover of e is s e = ∧{f ∈ T⎮f ≥ e},
and
the T-interior of e is eo = ∨{f ∈ T⎮f ≤ e}.
Clearly an idempotent e ∈ Δ belongs to T if and only if it equals either, and hence both, its T-
cover and its T-interior. We next state a pair of elementary lemmas, the first of which describes
an R-set in matrix form.
Lemma 6.1.8. Given e ∈ Δ and x ∈ Re = e + eR(1 – e), then e and x have the same
diagonal entries. Moreover, the nonzero diagonal entries of x occur only in those rows indexed
by supp(e) and in those columns with support indexed by supp(1 – e) . £
Lemma 6.1.9. Given e ∈T and j ∈supp(1 – e), e ∧ ej = 0 iff j ∈supp(1 – e)o . £
We now state our principal result for right-handed skew lattices in matrix rings.
Theorem 6.1.10. Let R be the ring of all n×n matrices over a (skew) field F, let Δn be
the lattice of all 0-1 diagonal matrices and let T be any sublattice of Δn containing at least the
zero and identity matrices, 0 and I. If the skew lattice S is the right extension of T in R, then for
each e ∈ S its D-class Se in S is described as follows:
i) All matrices in Se have the same diagonal as e.
ii) The nonzero, non-diagonal entries of any matrix in Se occur only in those
columns indexed by supp(1 – e).
iii) For j in supp(1 – e), the only positions in the jth column of a matrix in Se
that admit nonzero entries are given by supp(e ∧ ej ).
iv) For any j ∈ supp(1 – e), the jth column only has 0s when j ∈supp(1 – e)o .
v) No further restrictions are imposed on the matrices in Se.
Proof. Letting e ∈ T as stated, the class of matrices in Se is the intersection of classes of
matrices themselves obtained by appropriate juggling of the block designs given after Theorem
6.1.7. Thus to determine the matrices in Se, one need only discover which non-diagonal positions
can hold nonzero entries in these matrices since these positions can hold any member of F. Since
Se ⊆ Re, these “free” positions can only occur in rows indexed by supp(e) and columns indexed
by supp(1 – e). Thus (i) and (ii) are seen. Next let j ∈supp(1 – e) and let Γ(e, j) ⊆ supp(e) consist
of those jth column positions admitting non-0 entries.
Claim: Γ(e, j) = supp(e ∧ ej ) for all j ∈ supp(1 – e).
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In what follows, without loss of generality we assume that T is a sublattice of Δ such that
both matrices 0 and I lie in T. With this in mind, for each e ∈ Δ,
the T-cover of e is s e = ∧{f ∈ T⎮f ≥ e},
and
the T-interior of e is eo = ∨{f ∈ T⎮f ≤ e}.
Clearly an idempotent e ∈ Δ belongs to T if and only if it equals either, and hence both, its T-
cover and its T-interior. We next state a pair of elementary lemmas, the first of which describes
an R-set in matrix form.
Lemma 6.1.8. Given e ∈ Δ and x ∈ Re = e + eR(1 – e), then e and x have the same
diagonal entries. Moreover, the nonzero diagonal entries of x occur only in those rows indexed
by supp(e) and in those columns with support indexed by supp(1 – e) . £
Lemma 6.1.9. Given e ∈T and j ∈supp(1 – e), e ∧ ej = 0 iff j ∈supp(1 – e)o . £
We now state our principal result for right-handed skew lattices in matrix rings.
Theorem 6.1.10. Let R be the ring of all n×n matrices over a (skew) field F, let Δn be
the lattice of all 0-1 diagonal matrices and let T be any sublattice of Δn containing at least the
zero and identity matrices, 0 and I. If the skew lattice S is the right extension of T in R, then for
each e ∈ S its D-class Se in S is described as follows:
i) All matrices in Se have the same diagonal as e.
ii) The nonzero, non-diagonal entries of any matrix in Se occur only in those
columns indexed by supp(1 – e).
iii) For j in supp(1 – e), the only positions in the jth column of a matrix in Se
that admit nonzero entries are given by supp(e ∧ ej ).
iv) For any j ∈ supp(1 – e), the jth column only has 0s when j ∈supp(1 – e)o .
v) No further restrictions are imposed on the matrices in Se.
Proof. Letting e ∈ T as stated, the class of matrices in Se is the intersection of classes of
matrices themselves obtained by appropriate juggling of the block designs given after Theorem
6.1.7. Thus to determine the matrices in Se, one need only discover which non-diagonal positions
can hold nonzero entries in these matrices since these positions can hold any member of F. Since
Se ⊆ Re, these “free” positions can only occur in rows indexed by supp(e) and columns indexed
by supp(1 – e). Thus (i) and (ii) are seen. Next let j ∈supp(1 – e) and let Γ(e, j) ⊆ supp(e) consist
of those jth column positions admitting non-0 entries.
Claim: Γ(e, j) = supp(e ∧ ej ) for all j ∈ supp(1 – e).
216
