It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.
An example from [[linear algebra]] is the [[Matrix multiplication|multiplicative]]a [[monoid]]set of [[Real number|real]]-valued n-by-n square [[Matrix (mathematics)|matrices]]ofwith order ''n''the (calledmatrix-transpose as the [[full linear monoid]])involution. The [[Map (mathematics)|map]] which sends a matrix to its [[transpose]] is an involution because the transpose is well defined for any matrix and obeys the law {{nowrap|1=(''AB'')<sup>T</sup> = ''B''<sup>T</sup>''A''<sup>T</sup>}}, which has the same form of interaction with multiplication as taking inverses has in the [[general linear group]] (which is a subgroup of the full linear monoid). However, for an arbitrary matrix, ''AA''<sup>T</sup> does not equal the identity element (namely the [[diagonal matrix]]). Another example, coming from [[formal language]] theory, is the [[free semigroup]] generated by a [[nonempty set]] (an [[Alphabet (computer science)|alphabet]]), with string [[concatenation]] as the binary operation, and the involution being the map which [[String (computer science)#Reversal|reverse]]s the [[linear order]] of the letters in a string. A third example, from basic [[set theory]], is the set of all [[binary relation]]s between a set and itself, with the involution being the [[converse relation]], and the multiplication given by the usual [[composition of relations]].
Semigroups with involution appeared explicitly named in a 1953 paper of [[Viktor Wagner]] (in Russian) as result of his attempt to bridge the theory of semigroups with that of [[semiheap]]s.<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=265}}</ref>