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Idea
A ternary function which behaves as simultaneous actions on both the left and the right side:
Definition
Given a set and monoids and , a biaction is a ternary function such that
A single action monoid
-
for all ,
-
for all , , , , and ,
Given a set and a monoid , a -biaction is a ternary function such that
-
for all ,
-
for all , , , , and ,
Two action monoids
Given a set and monoids and , a --biaction is a ternary function such that
-
for all ,
-
for all , , , , and ,
Left and right actions
The left $M$-action? is defined as
for all and . It is a left action because
The right $N$-action? is defined as
for all and . It is a right action because
The left -action and right -action satisfy the following identity:
- for all , and , .
This is because when expanded out, the identity becomes:
See also