A structure theorem is provided for the free product
S
inv
T
S\,{\operatorname {inv}}\,T
of inverse semigroups
S
S
and
T
T
. Each element of
S
inv
T
S\,{\operatorname {inv}}\,T
is uniquely expressible in the form
ε
(
A
)
a
\varepsilon (A)a
, where
A
A
is a certain finite set of “left reduced” words and either
a
=
1
a = 1
or
a
=
a
1
⋯
a
m
a = {a_1} \cdots {a_m}
is a “reduced” word with
a
a
m
−
1
∈
A
aa_m^{ - 1} \in A
. (The word
a
1
⋯
a
m
{a_1} \cdots {a_m}
in
S
sgp
T
S\,{\operatorname {sgp}}\,T
is called reduced if no letter is idempotent, and left reduced if exactly
a
m
{a_m}
is idempotent; the notation
ε
(
A
)
\varepsilon (A)
stands for
Π
{
a
a
−
1
:
a
∈
A
}
\Pi \{ a{a^{ - 1}}:\,a \in A\}
.) Under a product remarkably similar to Scheiblich’s product for free inverse semigroups, the corresponding pairs
(
A
,
a
)
(A,\,a)
form an inverse semigroup isomorphic with
S
inv
T
S\,{\operatorname {inv}}\,T
. This description enables various properties of
S
inv
T
S\,{\operatorname {inv}}\,T
to be determined. For example
(
S
inv
T
)
∖
(
S
∪
T
)
(S\:{\operatorname {inv}}\:T)\backslash (S \cup T)
is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of
S
S
or
T
T
. If neither
S
S
nor
T
T
has a zero then
(
S
inv
T
)
(S\:{\operatorname {inv}}\:T)
is fundamental, but in general fundamentality itself is not preserved.