What is the name of an algebraic structure between semi-group and group?

Question

A group has an identity element and a an inverse element. A semi-group has neither. What is it called when a semi-group does have an identity element but no inversion?

The particular use-case is the "concatenation" operation on the set of all "strings" possible. Strings are a semi-group under concatenation:

• Closure: concat(string1, string2) is always another string
• Associativity: concat(string1, concat(string2, string3)) = concat(concat(string1, string2), string3)

And even though there is no inversion, there does exist an identity element: the empty string ("").

• concat(string1, "") = concat("", string1) = string1

Answer 1

Not only monoid is the standard term, but there is a tag monoid on this site with 679 questions as of today.

The example you describe is the free monoid on a set $A$, usually denoted by $A^*$, a frequently used object in theoretical computer science.

Answer 2

Apparently it's called a monoid, and in some cases a unitary or unital semi-group. I have to admit the latter sounds more accurate even though it is less used. Thanks to everyone who commented the answer!