Page 260 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 260
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
⎡G A1 Ak C ⎤2 ⎡G A1 Ak C⎤
⎢ D1 0 ⎥ ⎢ D1 0 ⎥
⎢0 ⎢ 0
B1 ⎥ B1 ⎥
⎢ ⎥ = ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ 0 0 Dk Bk ⎥ ⎢ 0 0 Dk Bk ⎥
⎣⎢ 0 0 0 H ⎦⎥ ⎣⎢ 0 0 0 H ⎦⎥
precisely when first G2 = G = 0 and H2 = H = 0 and the As, Bs, Cs and Ds behave as described in
Theorem 6.6.4. We have proved the following theorem.
Theorem 6.6.6. Let R be the set of all matrices of a Type (4) design in Mn(F) where:
1) The Ai, Bj and C blocks can be any matrix of the prescribed dimensions.
2) The Di-blocks are subject to the constraints of Theorem 6.6.4.
3) G and H belong to subrings Tʹ and Tʺ of Mp(F) and Mq(F) respectively, that
are maximal with respect to having no nonzero idempotent.
Then R is a maximal idempotent-closed subring of Mn(F). Its Type (3) matrix subring is a
maximal idempotent-closed and dominated subring of Mn(F).
The converse (a maximal idempotent-closed subring of Mn(F) containing a maximal
idempotent-closed and dominated subring of Mn(F), is simultaneously similar to a ring of the
above type) is also true. Its proof is given in Cvetko-Vah and Leech [2011].
Historical remarks
The results in Section 6.1 are from Leech’s initial paper on skew lattices [1989]. Those
in Sections 6.2 and 6.3 are from Cvetko-Vah and Leech’s 2008 paper, some of which generalized
results in two earlier papers of Cvetko-Vah ([2004] and ([2005a]. All results in Sections 6.4 and
6.5 are from the 2012 paper of Cvetko-Vah and Leech. Section 6.6 is based on earlier results of
Cvetko-Vah ([2005b] and [2007]).except for those mentioned at the end which are from their
2011 paper.
258
⎡G A1 Ak C ⎤2 ⎡G A1 Ak C⎤
⎢ D1 0 ⎥ ⎢ D1 0 ⎥
⎢0 ⎢ 0
B1 ⎥ B1 ⎥
⎢ ⎥ = ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ 0 0 Dk Bk ⎥ ⎢ 0 0 Dk Bk ⎥
⎣⎢ 0 0 0 H ⎦⎥ ⎣⎢ 0 0 0 H ⎦⎥
precisely when first G2 = G = 0 and H2 = H = 0 and the As, Bs, Cs and Ds behave as described in
Theorem 6.6.4. We have proved the following theorem.
Theorem 6.6.6. Let R be the set of all matrices of a Type (4) design in Mn(F) where:
1) The Ai, Bj and C blocks can be any matrix of the prescribed dimensions.
2) The Di-blocks are subject to the constraints of Theorem 6.6.4.
3) G and H belong to subrings Tʹ and Tʺ of Mp(F) and Mq(F) respectively, that
are maximal with respect to having no nonzero idempotent.
Then R is a maximal idempotent-closed subring of Mn(F). Its Type (3) matrix subring is a
maximal idempotent-closed and dominated subring of Mn(F).
The converse (a maximal idempotent-closed subring of Mn(F) containing a maximal
idempotent-closed and dominated subring of Mn(F), is simultaneously similar to a ring of the
above type) is also true. Its proof is given in Cvetko-Vah and Leech [2011].
Historical remarks
The results in Section 6.1 are from Leech’s initial paper on skew lattices [1989]. Those
in Sections 6.2 and 6.3 are from Cvetko-Vah and Leech’s 2008 paper, some of which generalized
results in two earlier papers of Cvetko-Vah ([2004] and ([2005a]. All results in Sections 6.4 and
6.5 are from the 2012 paper of Cvetko-Vah and Leech. Section 6.6 is based on earlier results of
Cvetko-Vah ([2005b] and [2007]).except for those mentioned at the end which are from their
2011 paper.
258
