Formalizer

\begin{definition} \label{def_inverse_semigroup_with_zero}
An \emph{inverse semigroup with zero} is an inverse semigroup that has
a zero element.
\end{definition}

\begin{definition} \label{def_inverse_semigroup_with_zero}
$S$ is an inverse semigroup with zero iff $S$ is an inverse semigroup and there exist $S_0, m, i, z$ such that $S = (S_0, m, i)$ and $z \in
S_0$ and for all $s \in S_0$ we have $m(z, s) = z$ and $m(s, z) = z$.
\end{definition}

\begin{definition} \label{def_idempotent_in_semigroup}
An \emph{idempotent} in a semigroup is an element $e$ satisfying $e^2=e$.
\end{definition}

\begin{definition} \label{def_idempotent_in_semigroup}
$e$ is an idempotent of $S$ iff $S$ is a semigroup and there exist $S_0, m$ such that $S = (S_0, m)$ and $e \in S_0$ and $m(e, e) = e$.
\end{definition}

\begin{definition} \label{def_join}
        In a poset $(P,\le)$, the \emph{join} of two elements $x,y$—when it
        exists—is their least upper bound, written $x\vee y$. More generally,
        for a subset $A\subseteq P$ the join $\bigvee A$ is the least element
        that lies above every member of $A$, provided such an element exists.
\end{definition}

\begin{definition} \label{def_join}
$z$ is the join of $A$ in $P$ with respect to $R$ iff for all $w \in R$ there exist $x, y \in P$ such that $w = (x, y)$ and for all $x \in
P$ $(x, x) \in R$ and for all $x, y \in P$ if $(x, y) \in R$ and $(y, x) \in R$ then $x = y$ and for all $x, y, z \in P$ if $(x, y) \in R$
and $(y, z) \in R$ then $(x, z) \in R$ and $A \subseteq P$ and $z \in P$ and for all $a \in A$ $(a, z) \in R$ and for all $y \in P$ if for
all $a \in A$ $(a, y) \in R$ then $(z, y) \in R$.
\end{definition}

\begin{definition} \label{def_compatible_subset}
        A subset $A$ of an inverse semigroup $S$ is
        \emph{compatible} if $s^{-1}t$ and $st^{-1}$ are idempotent for all
        $s\neq t$ in $A$.
\end{definition}

\begin{definition} \label{def_compatible_subset}
$A$ is a compatible subset of $S$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $A \subseteq
S_0$ and for all $s, t \in A$ if $s \neq t$ then $m(i(s), t)$ is an idempotent of $S$ and $m(s, i(t))$ is an idempotent of $S$.
\end{definition}

\begin{definition} \label{def_atom}
        Let $(P,\le)$ be a poset with least element $0$. An \emph{atom} is
        a minimal non‑zero element: $a\neq 0$ and
        $0< b\le a\Rightarrow b=a$.
\end{definition}

\begin{definition} \label{def_atom}
$a$ is an atom iff there exist $P, R, z$ such that $a \in P$ and for all $x \in P$ $(x, x) \in R$ and for all $x, y \in P$ if $(x, y) \in R$
and $(y, x) \in R$ then $x = y$ and for all $x, y, w \in P$ if $(x, y) \in R$ and $(y, w) \in R$ then $(x, w) \in R$ and $z \in P$ and for
all $x \in P$ $(z, x) \in R$ and $a \neq z$ and for all $b \in P$ if $(z, b) \in R$ and $(b, a) \in R$ and $b \neq z$ then $b = a$.
\end{definition}

\begin{definition} \label{def_groupoid}
        A \emph{groupoid} is a small category in which every morphism is
        invertible.
\end{definition}

\begin{definition} \label{def_groupoid}
$G$ is a groupoid iff there exist $O, M, s, t, c, i$ such that $G = (O, M, s, t, c, i)$ and $s \in \funs{M}{O}$ and $t \in \funs{M}{O}$ and
$c \in \funs{M \times M}{M}$ and $i \in \funs{O}{M}$ and for all $f \in M$ we have $s(i(s(f))) = s(f)$ and $t(i(s(f))) = s(f)$ and for all
$f, g \in M$ if $s(g) = t(f)$ then $s(c(f, g)) = s(f)$ and $t(c(f, g)) = t(g)$ and for all $f, g, h \in M$ if $s(h) = t(g)$ and $s(g) =
t(f)$ then $c(f, c(g, h)) = c(c(f, g), h)$ and for all $f \in M$ we have $c(f, i(s(f))) = f$ and $c(i(t(f)), f) = f$ and for all $f \in M$
there exists $g \in M$ such that $c(f, g) = i(t(f))$ and $c(g, f) = i(s(f))$.
\end{definition}

\begin{definition} \label{def_semigroup_ideal}
        For a semigroup $S$, a non‑empty subset $I\subseteq S$ is an
        \emph{ideal} if $SI\cup IS\subseteq I$; equivalently,
        $sIt\subseteq I$ for all $s,t\in S$.
\end{definition}

\begin{definition} \label{def_semigroup_ideal}
$I$ is an ideal of $S$ iff $S$ is a semigroup and there exist $S_0, m$ such that $S = (S_0, m)$ and $I \subseteq S_0$ and $I \neq \emptyset$
and $\{z \mid \exists s \in S_0 . \exists i \in I . z = m(s, i) \} \union \{z \mid \exists s \in S_0 . \exists i \in I . z = m(i, s) \}
\subseteq I$.
\end{definition}

\begin{definition} \label{def_semigroup_congruence}
        A \emph{congruence} on a semigroup $S$ is an equivalence relation
        $\rho$ on $S$ that is compatible with multiplication: whenever
        $s_1\,\rho\,t_1$ and $s_2\,\rho\,t_2$, one has
        $s_1s_2\,\rho\,t_1t_2$.
\end{definition}

\begin{definition} \label{def_semigroup_congruence}
$\rho$ is a congruence on $S$ iff $S$ is a semigroup and there exist $S_0, m$ such that $S = (S_0, m)$ and $\rho \subseteq S_0 \times S_0$
and for all $s \in S_0$ we have $(s, s) \in \rho$ and for all $s, t \in S_0$ if $(s, t) \in \rho$ then $(t, s) \in \rho$ and for all $s, t,
u \in S_0$ if $(s, t) \in \rho$ and $(t, u) \in \rho$ then $(s, u) \in \rho$ and for all $s_1, t_1, s_2, t_2 \in S_0$ if $(s_1, t_1) \in
\rho$ and $(s_2, t_2) \in \rho$ then $(m(s_1, s_2), m(t_1, t_2)) \in \rho$.
\end{definition}

\begin{definition} \label{def_maximal_subgroup}
        In a semigroup $S$, the set
        $$ H_e=\{s\in S: ss^{-1}=s^{-1}s=e\} $$
        associated to an idempotent $e$ is a subgroup of $S$, called the
        \emph{maximal subgroup at $e$}.
\end{definition}

\begin{definition} \label{def_maximal_subgroup}
$H$ is the maximal subgroup at $e$ in $S$ iff $S$ is an inverse semigroup and $e$ is an idempotent of $S$ and there exist $S_0, m, i$ such
that $S = (S_0, m, i)$ and $H = \{s \in S_0 \mid m(s, i(s)) = e \land m(i(s), s) = e \}$ and $e \in H$ and for all $a, b \in H$ we have
$m(a, b) \in H$ and for all $a \in H$ we have $i(a) \in H$.
\end{definition}

\begin{definition} \label{def_group_of_units}
The \emph{group of units} $\mathsf U(S)$ of a (unital) monoid $S$ is the subgroup of all invertible elements of $S$.
\end{definition}

\begin{definition} \label{def_group_of_units}
$\mathsf{U}(S) = \{s \in S_0 \mid m(s, i(s)) = e \land m(i(s), s) = e \}$.
\end{definition}

\begin{definition} \label{def_local_monoid}
If $ e $ is an idempotent in an inverse semigroup $ S $, then the subsemigroup $ eSe $ is called a \emph{local monoid}.
\end{definition}

\begin{definition} \label{def_local_monoid}
$M$ is a local monoid of $S$ iff $S$ is an inverse semigroup and there exist $S_0, m, i, e$ such that $S = (S_0, m, i)$ and $e$ is an
idempotent of $S$ and $M = (M_0, m)$ and $M_0 = \{m(e, m(s, e)) \mid s \in S_0 \}$ and for all $a, b \in M_0$ we have $m(a, b) \in M_0$.
\end{definition}

\begin{definition} \label{def_local_group_at_e}
The group of units of the local monoid $ eSe $ is denoted $ H_e $ and called the \emph{local group at $ e $}.
\end{definition}

\begin{definition} \label{def_local_group_at_e}
$H$ is the local group at $e$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $e$ is an
idempotent of $S$ and $H = \{s \in \{m(e, m(t, e)) \mid t \in S_0 \} \mid m(s, i(s)) = e \land m(i(s), s) = e \}$.
\end{definition}

\begin{definition} \label{def_domain}
For an element $ s $ in an inverse semigroup $ S $, the \emph{domain} of $ s $ is defined as the idempotent
$$ d(s)=s^{-1}s. $$
\end{definition}

\begin{definition} \label{def_domain}
$e$ is the domain of $s$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $s \in S_0$ and $e =
m(i(s), s)$.
\end{definition}

\begin{definition} \label{def_range}
For an element $ s $ in an inverse semigroup $ S $, the \emph{range} of $ s $ is defined as the idempotent
$$ r(s)=ss^{-1}. $$
\end{definition}

\begin{definition} \label{def_range}
$e$ is the range of $s$ in $S$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $s \in S_0$ and
$e = m(s, i(s))$.
\end{definition}

\begin{definition} \label{def_inverse_subsemigroup}
An \emph{inverse subsemigroup} of an inverse semigroup $T$ is a subsemigroup $S \subseteq T$ that is closed under the inversion map.
\end{definition}

\begin{definition} \label{def_inverse_subsemigroup}
$S$ is an inverse subsemigroup of $T$ iff $T$ is an inverse semigroup and there exist $T_0, m, i$ such that $T = (T_0, m, i)$ and $S
\subseteq T_0$ and for all $a, b \in S$ we have $m(a, b) \in S$ and for all $s \in S$ we have $i(s) \in S$.
\end{definition}

\begin{definition} \label{def_wide_inverse_subsemigroup}
An inverse subsemigroup $S$ of an inverse semigroup $T$ is called \emph{wide} if it contains all the idempotents of $T$.
\end{definition}

\begin{definition} \label{def_wide_inverse_subsemigroup}
$S$ is a wide inverse subsemigroup of $T$ iff $S$ is an inverse subsemigroup of $T$ and for all $e$ such that $e$ is an idempotent of $T$ we
have $e \in S$.
\end{definition}

\begin{definition} \label{def_clifford_semigroup}
A \emph{Clifford semigroup} is an inverse semigroup in which every idempotent is central (equivalently, $ d(s)= r(s)$ for all $s$).
\end{definition}

\begin{definition} \label{def_clifford_semigroup}
$S$ is a Clifford semigroup iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and for all $e$ such
that $e$ is an idempotent of $S$, for all $s \in S_0$ we have $m(e, s) = m(s, e)$.
\end{definition}

\begin{definition} \label{label_def_clifford_union_of_groups}
A semigroup $S$ is a \emph{union of groups} if
$$ S=\bigcup_{e\in \mathsf E(S)}H_e, $$
where $\mathsf E(S)$ is the set of idempotents of $S$, and each $H_e$ is a maximal subgroup associated to the idempotent $e$.
\end{definition}

\begin{definition} \label{label_def_clifford_union_of_groups}
$S$ is a union of groups iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and for all $s \in S_0$
there exists $e$ such that $e$ is an idempotent of $S$ and $m(s, i(s)) = e$ and $m(i(s), s) = e$.
\end{definition}

\begin{definition} \label{def_natural_partial_order}
        The \emph{natural partial order} on an inverse semigroup $S$ is
        $$ s \le t \quad\Longleftrightarrow\quad s=ts^{-1}s, $$
        equivalently $s=et$ or $s=tf$ for some idempotent $e,f$.
        \end{definition}

\begin{definition} \label{def_natural_partial_order}
$s$ is less than or equal to $t$ in $S$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $s, t
\in S_0$ and $s = m(t, m(i(s), s))$.
\end{definition}

\begin{definition} \label{def_inversible_semigroup_compatibility}
In an inverse semigroup $S$, two elements $s,t$ are \emph{compatible}, written $s\sim t$, if
$$ s^{-1}t\quad\text{and}\quad st^{-1} $$
are idempotents.
\end{definition}

\begin{definition} \label{def_inversible_semigroup_compatibility}
$s$ is compatible with $t$ in $S$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $s, t \in S_0$
and $m(i(s), t)$ is an idempotent of $S$ and $m(s, i(t))$ is an idempotent of $S$.
\end{definition}

\begin{definition} \label{def_orthogonal_inverse_semigroup_elements}
In an inverse semigroup with zero, two elements $ s,t $ are \emph{orthogonal}, written $ s \perp t $, if
$$ d(s)\, d(t) = 0 \quad \text{and} \quad r(s)\, r(t) = 0. $$
\end{definition}

\begin{definition} \label{def_orthogonal_inverse_semigroup_elements}
$s$ is orthogonal to $t$ iff $S$ is an inverse semigroup with zero and there exist $S_0, m, i, z$ such that $S = (S_0, m, i)$ and $s, t \in
S_0$ and there exist $d_s, d_t, r_s, r_t$ such that $d_s$ is the domain of $s$ and $d_t$ is the domain of $t$ and $r_s$ is the range of $s$
in $S$ and $r_t$ is the range of $t$ in $S$ and $m(d_s, d_t) = z$ and $m(r_s, r_t) = z$.
\end{definition}

\begin{definition} \label{def_partially_ordered_semigroup}
A monoid (or semigroup) endowed with a partial order $\le$ is a \emph{partially ordered semigroup} if
$$ a\le b,\;c\le d \quad\Longrightarrow\quad ac\le bd. $$
\end{definition}

\begin{definition} \label{def_partially_ordered_semigroup}
$S$ is a partially ordered semigroup iff there exist $S_0, m, R$ such that $S = (S_0, m, R)$ and $(S_0, m)$ is a semigroup and $R \subseteq
S_0 \times S_0$ and for all $a \in S_0$ we have $(a, a) \in R$ and for all $a, b \in S_0$ if $(a, b) \in R$ and $(b, a) \in R$ then $a = b$
and for all $a, b, c \in S_0$ if $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R$ and for all $a, b, c, d \in S_0$ if $(a, b) \in R$
and $(c, d) \in R$ then $(m(a, c), m(b, d)) \in R$.
\end{definition}

\begin{definition} \label{def_order_ideal}
A subset $ I $ of a poset $ (X, \le) $ is an \emph{order ideal} if whenever $ x \le y \in I $, then $ x \in I $.
\end{definition}

\begin{definition} \label{def_order_ideal}
$I$ is an order ideal iff there exist $X, R$ such that $I \subseteq X$ and for all $a \in X$ we have $(a, a) \in R$ and for all $a, b \in X$
if $(a, b) \in R$ and $(b, a) \in R$ then $a = b$ and for all $a, b, c \in X$ if $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R$ and
for all $x, y \in X$ if $(x, y) \in R$ and $y \in I$ then $x \in I$.
\end{definition}

\begin{definition} \label{def_principal_ideal}
The \emph{principal ideal} generated by an element $ y \in X $ is
$$ y^\downarrow = \{x \in X : x \le y \}. $$
\end{definition}

\begin{definition} \label{def_principal_ideal}
$y^\downarrow = \{x \in X \mid (x, y) \in R \}$.
\end{definition}

\begin{definition} \label{def_inverse_monoid_factorizable}
An inverse monoid $S$ is \emph{factorizable} if for each $s\in S$ there is a unit $g\in\mathsf U(S)$ with $s\le g$.
\end{definition}

\begin{definition} \label{def_inverse_monoid_factorizable}
$S$ is factorizable iff $S$ is an inverse monoid and there exist $S_0, m, i, e$ such that $S = (S_0, m, i)$ and for all $s \in S_0$ there
exists $g \in \mathsf{U}(S)$ such that $s$ is less than or equal to $g$ in $S$.
\end{definition}

\begin{definition} \label{def_minimum_group_congruence}
The \emph{minimum group congruence} $\sigma$ on an inverse semigroup $S$ is
$$ s\;\sigma\;t \quad\Longleftrightarrow\quad \exists u\le s,t. $$
Its quotient $S/\sigma$ is the maximal group homomorphic image.
\end{definition}

\begin{definition} \label{def_minimum_group_congruence}
$s \mathrel{\sigma} t$ iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and $s, t \in S_0$ and there
exists $u \in S_0$ such that $u$ is less than or equal to $s$ in $S$ and $u$ is less than or equal to $t$ in $S$.
\end{definition}

\begin{definition} \label{def_idempotent_separating}
A homomorphism $\theta: S \to T$ of inverse semigroups is \emph{idempotent-separating} if its restriction to idempotents is injective.
\end{definition}

\begin{definition} \label{def_idempotent_separating}
$\theta$ is an idempotent-separating homomorphism from $S$ to $T$ iff $S$ is an inverse semigroup and $T$ is an inverse semigroup and there
exist $S_0, m_S, i_S$ such that $S = (S_0, m_S, i_S)$ and there exist $T_0, m_T, i_T$ such that $T = (T_0, m_T, i_T)$ and $\theta \in
\funs{S_0}{T_0}$ and for all $a, b \in S_0$ we have $\theta(m_S(a, b)) = m_T(\theta(a), \theta(b))$ and for all $a \in S_0$ we have
$\theta(i_S(a)) = i_T(\theta(a))$ and for all $e_1, e_2 \in S_0$ if $e_1$ is an idempotent of $S$ and $e_2$ is an idempotent of $S$ and
$\theta(e_1) = \theta(e_2)$ then $e_1 = e_2$.
\end{definition}

\begin{definition} \label{def_idempotent_pure}
A homomorphism $\theta: S \to T$ of inverse semigroups is \emph{idempotent-pure} if $\theta(s)$ idempotent implies $s$ is idempotent.
\end{definition}

\begin{definition} \label{def_idempotent_pure}
$\theta$ is an idempotent-pure homomorphism from $S$ to $T$ iff $S$ is an inverse semigroup and $T$ is an inverse semigroup and there exist
$S_0, m, i$ such that $S = (S_0, m, i)$ and there exist $T_0, n, j$ such that $T = (T_0, n, j)$ and $\theta \in \funs{S_0}{T_0}$ and for all
$a, b \in S_0$ we have $\theta(m(a, b)) = n(\theta(a), \theta(b))$ and for all $a \in S_0$ we have $\theta(i(a)) = j(\theta(a))$ and for all
$s \in S_0$ if $\theta(s)$ is an idempotent of $T$ then $s$ is an idempotent of $S$.
\end{definition}

\begin{definition} \label{def_e_unitary_inverse_semigroup}
An inverse semigroup $S$ is \emph{$E$-unitary} if $e \le s$ with $e^2 = e \neq 0$ forces $s$ to be an idempotent.
Similarly, $E^*$-unitary if $0 \neq e \le s$ with $e$ idempotent implies $s$ is idempotent.
\end{definition}

\begin{definition} \label{def_e_unitary_inverse_semigroup}
$S$ is E-star-unitary iff $S$ is an inverse semigroup with zero and there exist $S_0, m, i, z$ such that $S = (S_0, m, i)$ and for all $e, s
\in S_0$ if $z \neq e$ and $e$ is an idempotent of $S$ and $e$ is less than or equal to $s$ in $S$ then $s$ is an idempotent of $S$.
\end{definition}

\begin{definition} \label{def_zero_disjunctive_meet_semilattice}
A \emph{$0$-disjunctive} meet-semilattice $E$ with zero is one in which for every $0 \neq f < e$, there is $0 \neq g \leq e$ with $fg = 0$.
\end{definition}

\begin{definition} \label{def_zero_disjunctive_meet_semilattice}
$E$ is a zero-disjunctive meet-semilattice with zero iff $E$ is a semigroup and there exist $E_0, m$ such that $E = (E_0, m)$ and for all
$a, b \in E_0$ we have $m(a, b) = m(b, a)$ and for all $a \in E_0$ we have $m(a, a) = a$ and there exists $z \in E_0$ such that for all $a
\in E_0$ we have $m(a, z) = z$ and $m(z, a) = z$ and for all $f, e \in E_0$ if $z \neq f$ and $m(f, e) = f$ and $f \neq e$ then there exists
$g \in E_0$ such that $z \neq g$ and $m(g, e) = g$ and $m(f, g) = z$.
\end{definition}

\begin{definition} \label{def_infinitesimal_in_meet_semilattice}
In a meet‐semilattice $E$, a nonzero $a\in E$ is an \emph{infinitesimal} if $a^2=0$.
\end{definition}

\begin{definition} \label{def_infinitesimal_in_meet_semilattice}
$a$ is an infinitesimal of $E$ iff $E$ is a semigroup and there exist $E_0, m$ such that $E = (E_0, m)$ and for all $a, b \in E_0$ we have
$m(a, b) = m(b, a)$ and for all $a \in E_0$ we have $m(a, a) = a$ and there exists $z \in E_0$ such that for all $b \in E_0$ we have $m(b,
z) = z$ and $m(z, b) = z$ and $a \in E_0$ and $a \neq z$ and $m(a, a) = z$.
\end{definition}

\begin{definition} \label{def_f_inverse}
An inverse monoid $S$ is \emph{$F$-inverse} if each $\sigma$-class contains a greatest element.
\end{definition}

\begin{definition} \label{def_f_inverse}
$S$ is $F$-inverse iff $S$ is an inverse monoid and there exist $S_0, m, i, e$ such that $S = (S_0, m, i)$ and there exists a congruence $C$
on $S$ such that for all $s \in S_0$ there exists $g \in S_0$ such that $(s, g) \in C$ and for all $t \in S_0$ if $(s, t) \in C$ then $t$ is
less than or equal to $g$ in $S$.
\end{definition}

\begin{definition} \label{def_pseudogroup}
A \emph{pseudogroup} is an inverse semigroup in which all compatible subsets have joins, and multiplication distributes over those joins.
\end{definition}

\begin{definition} \label{def_pseudogroup}
$P$ is a pseudogroup iff $P$ is an inverse semigroup and there exist $P_0, m, i$ such that $P = (P_0, m, i)$ and for all $A$ such that $A$
is a compatible subset of $P$ there exists $s \in P_0$ such that for all $a \in A$ $a$ is less than or equal to $s$ in $P$ and for all $t
\in P_0$ if for all $a \in A$ $a$ is less than or equal to $t$ in $P$ then $s$ is less than or equal to $t$ in $P$ and for all $t \in P_0$
for all $a \in A$ $m(t, a)$ is less than or equal to $m(t, s)$ in $P$ and for all $u \in P_0$ if for all $a \in A$ $m(t, a)$ is less than or
equal to $u$ in $P$ then $m(t, s)$ is less than or equal to $u$ in $P$ and for all $a \in A$ $m(a, t)$ is less than or equal to $m(s, t)$ in
$P$ and for all $u \in P_0$ if for all $a \in A$ $m(a, t)$ is less than or equal to $u$ in $P$ then $m(s, t)$ is less than or equal to $u$
in $P$.
\end{definition}

\begin{definition} \label{def_distributive_inverse_monoid}
A \emph{distributive inverse monoid} is one with all finite joins and in which multiplication distributes over them.
\end{definition}

\begin{definition} \label{def_distributive_inverse_monoid}
$M$ is a distributive inverse monoid iff $M$ is an inverse monoid and for all $a, b \in M_0$ there exists $s \in M_0$ such that $a$ is less
than or equal to $s$ in $M$ and $b$ is less than or equal to $s$ in $M$ and for all $t \in M_0$ if $a$ is less than or equal to $t$ in $M$
and $b$ is less than or equal to $t$ in $M$ then $s$ is less than or equal to $t$ in $M$ and for all $c \in M_0$ there exists $u \in M_0$
such that $m(c, a)$ is less than or equal to $u$ in $M$ and $m(c, b)$ is less than or equal to $u$ in $M$ and for all $v \in M_0$ if $m(c,
a)$ is less than or equal to $v$ in $M$ and $m(c, b)$ is less than or equal to $v$ in $M$ then $u$ is less than or equal to $v$ in $M$ and
$m(c, s) = u$ and there exists $w \in M_0$ such that $m(a, c)$ is less than or equal to $w$ in $M$ and $m(b, c)$ is less than or equal to
$w$ in $M$ and for all $x \in M_0$ if $m(a, c)$ is less than or equal to $x$ in $M$ and $m(b, c)$ is less than or equal to $x$ in $M$ then
$w$ is less than or equal to $x$ in $M$ and $m(s, c) = w$.
\end{definition}

\begin{definition} \label{def_boolean_algebra}
        A \emph{Boolean algebra} is a distributive lattice $(B,\wedge,\vee)$
        with least element $0$ and greatest element $1$ such that every
        $b\in B$ has a \emph{complement} $\bar b$ satisfying
        $b\wedge\bar b=0$ and $b\vee\bar b=1$.
\end{definition}

\begin{definition} \label{def_boolean_algebra}
$B$ is a Boolean algebra iff there exist $B_0, m, j, z, o, c$ such that $B = (B_0, m, j, z, o, c)$ and $m \in \funs{B_0 \times B_0}{B_0}$
and $j \in \funs{B_0 \times B_0}{B_0}$ and $z \in B_0$ and $o \in B_0$ and $c \in \funs{B_0}{B_0}$ and for all $a, b \in B_0$ we have $m(a,
b) = m(b, a)$ and $j(a, b) = j(b, a)$ and for all $a, b, c \in B_0$ we have $m(a, m(b, c)) = m(m(a, b), c)$ and $j(a, j(b, c)) = j(j(a, b),
c)$ and for all $a, b \in B_0$ we have $m(a, j(a, b)) = a$ and $j(a, m(a, b)) = a$ and for all $a, b, c \in B_0$ we have $m(a, j(b, c)) =
j(m(a, b), m(a, c))$ and $j(a, m(b, c)) = m(j(a, b), j(a, c))$ and for all $b \in B_0$ we have $m(b, z) = z$ and $j(b, z) = b$ and $m(b, o)
= b$ and $j(b, o) = o$ and for all $b \in B_0$ we have $m(b, c(b)) = z$ and $j(b, c(b)) = o$.
\end{definition}

\begin{definition} \label{def_boolean_inverse_monoid}
A \emph{Boolean inverse monoid} is a distributive inverse monoid whose semilattice of idempotents is a Boolean algebra.
\end{definition}

\begin{definition} \label{def_boolean_inverse_monoid}
$M$ is a Boolean inverse monoid iff $M$ is an inverse monoid and there exist $M_0, m, i$ such that $M = (M_0, m, i)$ and for all $a, b, c
\in M_0$ if $m(a, a) = a$ and $m(b, b) = b$ and $m(c, c) = c$ then $m(a, m(b, c)) = m(m(a, b), m(a, c))$ and there exist $z, u \in M_0$ such
that $m(z, z) = z$ and $m(u, u) = u$ and for all $e \in M_0$ if $m(e, e) = e$ then $m(e, u) = e$ and $m(e, z) = z$ and for all $e \in M_0$
if $m(e, e) = e$ then there exists $f \in M_0$ such that $m(f, f) = f$ and $m(e, f) = z$ and $m(m(e, f), u) = u$.
\end{definition}

\begin{definition} \label{def_fixed_point_operator}
A \emph{fixed‐point operator} on an inverse monoid $S$ is a map $\phi\colon S\to\mathsf E(S)$ with
$$ \phi(a)\le a,\quad e\le a,\ e^2=e\implies e\le \phi(a), $$
together with the identities $\phi(ae)=\phi(a)e$ and $\phi(ea)=e\phi(a)$.
\end{definition}

\begin{definition} \label{def_fixed_point_operator}
$\phi$ is a fixed-point operator on $S$ iff $S$ is an inverse monoid and there exist $S_0, m, i, e$ such that $S = (S_0, m, i)$ and $\phi
\in \funs{S_0}{\{f \in S_0 \mid m(f, f) = f\}}$ and for all $a \in S_0$ we have $\phi(a)$ is less than or equal to $a$ in $S$ and for all $e
\in S_0$ if $m(e, e) = e$ and $e$ is less than or equal to $a$ in $S$ then $e$ is less than or equal to $\phi(a)$ in $S$ and for all $a, e
\in S_0$ we have $\phi(m(a, e)) = m(\phi(a), e)$ and $\phi(m(e, a)) = m(e, \phi(a))$.
\end{definition}

\begin{definition} \label{def_relative_complement_boolean_inverse_monoid}
In a Boolean inverse monoid $S$, for $y\le x$ the \emph{relative complement} is
$$ x\setminus y \;=\;x\;\overline{d(y)}. $$
\end{definition}

\begin{definition} \label{def_relative_complement_boolean_inverse_monoid}
$z$ is the relative complement of $y$ with respect to $x$ in $S$ iff $S$ is a Boolean inverse monoid and there exist $S_0, m, i$ such that
$S = (S_0, m, i)$ and $y$ is less than or equal to $x$ in $S$ and $z = m(x, \neg{m(i(y), y)})$.
\end{definition}

\begin{definition} \label{def_underlying_groupoid_of_inverse_semigroup}
The \emph{underlying groupoid} of an inverse semigroup $S$ is the category obtained by allowing the (total) product $st$ only
when $ d(s)= r(t)$.
\end{definition}

\begin{definition} \label{def_underlying_groupoid_of_inverse_semigroup}
$G$ is the underlying groupoid of $S$ iff $S$ is an inverse semigroup and there exist $S_0, m, k$ such that $S = (S_0, m, k)$ and there
exist $O, M, s, t, c, i$ such that $G = (O, M, s, t, c, i)$ and $O = \{e \in S_0 \mid m(e, e) = e \}$ and $M = S_0$ and $s \in \funs{M}{O}$
and for all $f \in M$ we have $s(f) = m(k(f), f)$ and $t \in \funs{M}{O}$ and for all $f \in M$ we have $t(f) = m(f, k(f))$ and $c \in
\funs{M \times M}{M}$ and for all $f, g \in M$ if $s(f) = t(g)$ then $c(f, g) = m(f, g)$ and $i \in \funs{O}{M}$ and for all $e \in O$ we
have $i(e) = e$.
\end{definition}

\begin{definition} \label{def_inverse_semigroup_atomic_groupoid}
In an inverse semigroup $S$, the \emph{atomic groupoid} $\mathsf A(S)$ consists of those atoms—minimal nonzero elements—of $S$, with the
restricted product.
\end{definition}

\begin{definition} \label{def_inverse_semigroup_atomic_groupoid}
$G$ is the atomic groupoid of $S$ iff $S$ is an inverse semigroup with zero and there exist $S_0, m, i, z$ such that $S = (S_0, m, i, z)$
and there exist $O, M, s, t, c, j$ such that $G = (O, M, s, t, c, j)$ and $O = \{e \in S_0 \mid \text{$e$ is an idempotent of $S$ and $e$ is
an atom}\}$ and $M = \{a \in S_0 \mid \text{$a$ is an atom}\}$ and $s \in \funs{M}{O}$ and for all $a \in M$ $s(a) = m(i(a), a)$ and $t \in
\funs{M}{O}$ and for all $a \in M$ $t(a) = m(a, i(a))$ and $c \in \funs{M \times M}{M}$ and for all $a, b \in M$ if $s(a) = t(b)$ then $c(a,
b) = m(a, b)$ and $j \in \funs{O}{M}$ and for all $e \in O$ $j(e) = e$ and $G$ is a groupoid.
\end{definition}

\begin{definition} \label{def_fundamental_inverse_semigroup}
A \emph{fundamental} inverse semigroup is one whose only elements commuting with all idempotents are idempotents themselves;
equivalently its maximum idempotent‐separating congruence $\mu$ is trivial.
\end{definition}

\begin{definition} \label{def_fundamental_inverse_semigroup}
$S$ is a fundamental inverse semigroup iff $S$ is an inverse semigroup and there exist $S_0, m, i$ such that $S = (S_0, m, i)$ and for all
$s \in S_0$ if for all $e \in S_0$ such that $e$ is an idempotent of $S$ we have $m(s, e) = m(e, s)$ then $s$ is an idempotent of $S$.
\end{definition}

\begin{definition} \label{def_inverse_semigroup_zero_0_simple}
An inverse semigroup with zero is \emph{$0$-simple} if its only ideals are $\{0\}$ and itself.
\end{definition}

\begin{definition} \label{def_inverse_semigroup_zero_0_simple}
$S$ is zero-simple iff $S$ is an inverse semigroup with zero and for all $I$ if $I$ is an ideal of $S$ then $I = \{z\}$ or $I = S_0$.
\end{definition}

\begin{definition} \label{def_annihilator_congruence}
On an inverse semigroup with zero, the \emph{annihilator congruence} $\xi$ is given by
$$ s\;\xi\;t \quad\Longleftrightarrow\quad \forall a,b:\;asb=0\iff atb=0. $$
It is the largest $0$-restricted congruence.
\end{definition}

\begin{definition} \label{def_annihilator_congruence}
$\rho$ is the annihilator congruence on $S$ iff $S$ is an inverse semigroup with zero and there exist $S_0, m, i, z$ such that $S = (S_0, m,
i)$ and $\rho$ is a congruence on $S$ and for all $s, t \in S_0$ we have $(s, t) \in \rho$ iff for all $a, b \in S_0$, $m(m(a, s), b) = z$
$\leftrightarrow$ <<< $m(m(a, t), b) = z$.
\end{definition}

\begin{definition} \label{def_congruence_free_inverse_semigroup_with_zero}
An inverse semigroup with zero is \emph{congruence‐free} if its only congruences are equality and the universal relation.
\end{definition}

\begin{definition} \label{def_congruence_free_inverse_semigroup_with_zero}
$S$ is congruence-free iff $S$ is an inverse semigroup with zero and there exist $S_0, m, i, z$ such that $S = (S_0, m, i)$ and $z \in S_0$
and for all $s \in S_0$ we have $m(z, s) = z$ and $m(s, z) = z$ and for all $\rho$ if $\rho$ is a congruence on $S$ then $\rho = \{(s, s)
\mid s \in S_0 \}$ or $\rho = S_0 \times S_0$.
\end{definition}