does not matter, i.e. any order of deletions produces the same result. (Otherwise put, these rules define a confluent rewriting system.) Equivalently, a free group is constructed from a free monoid with involution by taking the quotient of the latter by the congruence , which is sometimes called the Dyck congruence—in a certain sense it generalizes Dyck language to multiple kinds of "parentheses" However simplification in the Dyck congruence takes place regardless of order. For example, if ")" is the inverse of "(", then ; the one-sided congruence that appears in the Dyck language proper , which instantiates only to is (perhaps confusingly) called the Shamir congruence. The quotient of a free monoid with involution by the Shamir congruence is not a group, but a monoid ; nevertheless it has been called the free half group by its first discoverer—Eli Shamir—although more recently it has been called the involutive monoid generated by X. (This latter choice of terminology conflicts however with the use of "involutive" to denote any semigroup with involution—a practice also encountered in the literature.) Baer *-semigroups A Baer *-semigroup is a *-semigroup with (two-sided) zero in which the right annihilator of every element coincides with the right ideal of some projection; this property is expressed formally as: for all x ∈ S there exists a projection e such that { y ∈ S | xy = 0 } = eS. The projection e is in fact uniquely determined by x. More recently, Baer *-semigroups have been also called Foulis semigroups, after David James Foulis who studied them in depth. Examples and applications The set of all binary relations on a set (from example 5) is a Baer *-semigroup. Baer *-semigroups are also encountered in quantum mechanics, in particular as the multiplicative semigroups of Baer *-rings. If H is a Hilbert space, then the multiplicative semigroup of all bounded operators on H is a Baer *-semigroup. The involution in this case maps an operator to its adjoint. Baer *-semigroup allow the coordinatization of orthomodular lattices. See also Dagger category (aka category with involution) — generalizes the notion *-algebra Special classes of semigroups Notes References Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". World Scientific D J Foulis (1958). Involution Semigroups, PhD Thesis, Tulane University, New Orleans, LA. Publications of D.J. Foulis (Accessed on 5 May 2009) W.D. Munn, Special Involutions, in A.H. Clifford, K.H. Hofmann,