Page 87 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 87
II: Skew Lattices
Theorem 2.6.14. Every right-handed, strongly distributive skew lattice can be embedded
in a power of 3R. Equivalently, every such skew lattice can be embedded in some PR(A, {1, 2}).
Thus, every right-handed strongly distributive skew lattice S is isomorphic to a ring of partial
functions. If S/D is finite, then S is isomorphic to a full ring of partial functions. £
Example 2.6.3 (Example 2.3.4 revisited). This matrix example is isomorphic to the
direct product of the primitive algebras (F × F)0 where F × F is the rectangular algebra, that is, it
is isomorphic to the power [(F × F)0]n. £
As indicated above, a natural continuation of these ideas is given in Chapter 4, and in
particular, in the first section.
85
Theorem 2.6.14. Every right-handed, strongly distributive skew lattice can be embedded
in a power of 3R. Equivalently, every such skew lattice can be embedded in some PR(A, {1, 2}).
Thus, every right-handed strongly distributive skew lattice S is isomorphic to a ring of partial
functions. If S/D is finite, then S is isomorphic to a full ring of partial functions. £
Example 2.6.3 (Example 2.3.4 revisited). This matrix example is isomorphic to the
direct product of the primitive algebras (F × F)0 where F × F is the rectangular algebra, that is, it
is isomorphic to the power [(F × F)0]n. £
As indicated above, a natural continuation of these ideas is given in Chapter 4, and in
particular, in the first section.
85
