Is there a standard name for this semigroup?

Question

Given a semigroup $X,$ we can form a new semigroup $Y$ by asserting that:

• the carrier of $Y$ is the set $X^2$, and
• the law of composition in $Y$ is given by $(a,b)(a',b')=(aa',b'b).$

Finally, define that the left-action of an element of $Y$ on an element of $X$ satisfies $$(a,b)x=axb.$$

Under these definitions, it is easy to see that, for all $y,y' \in Y$ and all $x \in X$ we have $$(yy')x=y(y'x).$$

Is there a standard name for this construction? And where can I learn more about it?

Answer 1

$Y$ is the direct product of $X$ with its opposite, the semigroup denoted by $X^{\text{op}}$: $$Y = X \times X^{\text{op}}$$

If $X$ is a semigroup, we can define a semigroup $X^{\text{op}}$ such that $X, X^{\text{op}}$ are opposite semigroups: As Jack notes in the comments, a left action of $X^{\text{op}}$ is the same as a right action of $X$.

For some nicely compiled notes from Pete L. Clark, see Introduction to semigroups and monoids. You might find this post helpful, too.