| | 5 Kleene and Omega
Algebra
Let denote ,
is an idempotent left semiring iff (for )
is a commutative monoid , i.e.,
is a monoid, i.e.,
implies where iff
is a Boolean idempotent left semiring iff
is an idempotent left semiring
(Huntington equation)
As usual we have the following
(greatest element w.r.t. , i.e. )
(smallest element w.r.t. , i.e., )
is a Boolean strong left (lazy) Kleene algebra iff
is a Boolean idempotent left semiring
implies
implies 
is a Boolean left (lazy) omega algebra iff
is a Boolean strong left (lazy) Kleene algebra
implies 
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