Commutator subspace

In Boolean algebra (structure), the inclusion relation $a\le b$ is defined as $a{b}^{\prime }=0$ and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order.

The inclusion relation $a<b$ can be expressed in many ways:

• $a<b$
• $a{b}^{\prime }=0$
• $a^{\prime }+b=1$
• $b^{\prime }<a^{\prime }$
• $a+b=b$
• $ab=a$

The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }.

Some useful properties of the inclusion relation are:

• $a\le a+b$
• $ab\le a$

The inclusion relation may be used to define Boolean intervals such that $a\le x\le b$ A Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.

References

• Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 52