# The intersection of *-semigroups with I-semigroups is the class of inverse semigroups?

Howie in his Fundamentals of Semigroup Theory, 2nd ed., p. 103 writes

The class of U-semigroups for which the unary operation satisfies the conditions both for a *-semigroup and for an I-semigroup is in fact the class of inverse semigroups, which will be the subject of Chapter 5.

where

• U-semigroups are defined as having a unary operation satisfying $x^{**}=x$,
• I-semigroups are U-semigroups in which $xx^*x = x$ also holds,
• *-semigroups are U-semigroups in which $(xy)^* = y^*x^*$.

But there's a class of semigroups (albeit not discussed in Howie) introduced by Nordahl and Scheiblich, which they called regular *-semigroups that seems to be the proper intersection of I-semigroups and *-semigroups. They give the following example of a regular *-semigroup that's not an inverse semigroup: given a set $A$ form a semigroup on $A × A$ with the semigroup product defined by $(a, b)(c, d) = (a, d)$ and the involution being $(a, b)^* = (b, a)$. It's utterly trivial to verify that this semigroup satisfies all three of the above equations so it is both a I-semigroup and *-semigroup.

To see that's not an inverse semigroup, the equation [which actually turns out to be an identity]: $(a, b)(p, q)(a, b) = (a, q)(a, b) = (a, b)$ with unknown $(p, q)$ accepts as solution any element of the semigroup. And any solution also verifies the "dual" equation/identity $(p, q)(a, b)(p, q) = (p, q)$. So this semigroup is a regular semigroup that is as far from an inverse semigroup as possible (in that class inverses must be unique), well, unless $A$ has single element.

So, is the statement I quoted from Howie's book simply erroneous or should it be interpreted in some way that eludes me? (There doesn't seem to be an errata for the book, by the way.)

In Chapter 5 [p. 145], Howie defines an inverse semigroup as an I-semigroup $S$ such that, additionally, for all $x, y \in S$, $xx^*yy^*=yy^*xx^*$. Howie then proves correctly that this definition is equivalent with the [more typical] definition that every element has a unique inverse [in Theorem 5.1.1(4)]. He then proves [Proposition 5.1.2(1)] that in an inverse semigroup $(xy)^* = y^*x^*$ also holds, so an inverse semigroup is also a *-semigroup.

Howie doesn't attempt however to back up his claim from p. 103 that a semigroup that is both a *-semigroup and a I-semigroup is inverse, and it's easy to see that in an I-semigroup that's also a *-semigroup, $xx^*yy^*=yy^*xx^*$ may not hold: in the semigroup example [rectangular band] of Nordahl and Scheiblich: $(a,b)(b,a)(c,d)(d,c) = (a,c)$ but $(c,d)(d,c)(a,b)(b,a) = (c,a)$, so these aren't equal unless the semigroup is trivial.

So there no surprises in Chapter 5, i.e. the problematic statement is localized on p. 103.

I think this is a mistake, there are several well-known and much studied classes of regular *-semigroups. At the end of page 102, Howie even writes so "$a^{-1}$ is an inverse of $a$", not "$a^{-1}$ is the unique inverse of $a$". Using $a^{-1}$ to mean "an inverse" rather than "the unique inverse" could easily lead to confusion.