--- title: Boolean Restriction Semigroups url: https://www.emergentmind.com/topics/boolean-restriction-semigroup type: topic --- # Boolean Restriction Semigroups A Boolean restriction semigroup is an algebraic structure that generalizes both inverse semigroups and Boolean algebras, encapsulating the one-sided logic of partial symmetries. Formally, a Boolean restriction semigroup is a restriction semigroup $(S,\cdot,{}^*)$ in which the set of projections $P(S)=\{e\in S:e^*=e\}$ forms a generalized Boolean algebra, and which admits finite joins of compatible elements distributing over multiplication. Such semigroups arise as the algebraic counterparts to ample topological categories (étale categories), and have deep connections to noncommutative Stone duality, groupoid theory, and operator algebras. ## 1. Axiomatic and Structural Foundations A restriction semigroup $(S,\cdot,{}^*)$ is a semigroup equipped with a unary operation (domain operation) $s^*$ satisfying for all $x,y\in S$: \[ \begin{aligned} &x\,x^* \;=\; x, &&(R1)\\ &x^*\,y^* \;=\; (y\,x)^*, &&(R2)\\ &(x\,y)^* \;=\; (x^*\,y)^*, &&(R3)\\ &x^*\,y \;=\; y\,(x\,y)^*. &&(R4) \end{aligned} \] The projections $P(S)$ under multiplication form a commutative semilattice. The **natural partial order** is $s\le t \iff s=t\,s^*\iff s=e t$ for some $e\in P(S)$. Compatibility of $s,t\in S$ is defined as $s\,t^* = t\,s^*$; such elements admit joins $s\vee t$. A Boolean restriction semigroup additionally satisfies: - (BR1): If $s\sim t$, then $s\vee t$ exists in $S$. - (BR2): $P(S)$ is a generalized Boolean algebra (distributive, complemented, with $0$). - (BR3): Multiplication distributes over joins of compatible pairs: $(s\vee t)u = su \vee tu$ for all $u\in S$. If the semigroup is also a monoid and all elements are generated under compatible joins from partial units, the structure is called **étale Boolean restriction monoid**, which is categorically equivalent to the class of Boolean inverse monoids [2404.08606]. ## 2. Examples and Constructions ### 2.1 Partial Maps The semigroup $\mathrm{PT}(X)$ of all partial self-maps of a set $X$, with $f^*=\operatorname{id}_{\operatorname{dom}(f)}$, forms a Boolean restriction monoid. The projections are identity maps on subsets of $X$, yielding a Boolean algebra. Compatible partial maps (agreeing on domain overlap) have their join as union. ### 2.2 Inverse and Binary-Relation Cases If $S$ is an inverse semigroup, $s^*=s^{-1}s$ provides the Boolean restriction structure, and all further Boolean structure is classical. In the semigroup of binary relations $\mathcal B(X)$, one defines operations $r^*$ and $r^+$, but $\mathcal B(X)$ is only a restriction semigroup for $|X|\le1$. ### 2.3 Thompson and Cuntz Monoids Given the polycyclic inverse monoid $P_n$ and its Boolean completion $C_n=P_n^\vee$ (the Cuntz inverse monoid), one constructs $H_n=\mathsf{Etale}(C_n)$ as the étale Boolean right-restriction monoid companion to $C_n$. The group of units of $H_n$ recovers Thompson's group $G_{n,1}$ [2404.08606]. ## 3. Universal and Categorical Correspondences ### 3.1 Universal Étale Category Every restriction semigroup $S$ with local units has an associated universal étale category $\mathscr{C}(S)$, the category of germs of the spectral action on the character space of $P(S)$. Concretely, with $X=\widehat{P(S)}$ the spectrum (set of nonzero semilattice homomorphisms), the *spectral action* is defined by partial homeomorphisms $\beta_s\colon D_{s^*}\to D_{s^*s^*}$, and the arrows are germs $[s,\varphi]$ for $\varphi\in D_{s^*}$. The set of compact slices $C(S)^{a}$ forms a Boolean restriction semigroup, and $S$ embeds into $C(S)^a$ via $s\mapsto (s,D_{s^*})$ [2511.03869]. ### 3.2 Equivalences of Categories A central result is the equivalence: \[ \left\{ \text{Boolean restriction semigroups} \right\} \simeq \left\{ \text{ample categories} \right\} \] where slices in the category correspond to elements of the semigroup, and conversely, from a preBoolean restriction semigroup with local units, one constructs its germ category, forming an ample category with structure matching the semigroup [2410.22181]. This equivalence extends to: - Boolean range semigroups $\simeq$ strongly ample categories (open range map), - Étale Boolean range semigroups $\simeq$ biample (two-sided étale) categories, - Boolean birestriction semigroups $\simeq$ biample categories, - Boolean inverse semigroups $\simeq$ ample groupoids. ## 4. Boolean Restriction Semigroups and Inverse Theory Boolean restriction semigroups generalize the theory of inverse semigroups, providing a precise algebraic framework for partial symmetries beyond the invertible case. The connection is made via the submonoid of **partial units**: \[ \mathsf{Inv}(S) = \{\, a\in S \mid \exists b:\; b\,a=a^*,\ a\,b=b^*\,\} \] This set is an inverse submonoid with the same idempotent structure as $S$. Étale Boolean right-restriction monoids are generated by finite (left-)compatible joins of their partial units, and there is a categorical equivalence: \[ \mathbf{EtaleRest} \simeq \mathbf{BoolInv} \] where the functor $S \mapsto \mathsf{Inv}(S)$ is an equivalence of categories [2404.08606]. ## 5. Topological, Algebraic, and Duality Aspects The core of the theory relates Boolean restriction semigroups to ample topological categories (étale categories), where slices correspond to elements, and arrows correspond to partial symmetries in the operator-theoretic or dynamical picture. The convolution algebras of compact slices generalize Steinberg algebras: \[ (\chi_{U}*\chi_{V})(x) = \sum_{y z = x} \chi_U(y)\chi_V(z) = \chi_{U\cdot V}(x) \] with an induced isomorphism $K[S] \cong K C(S)$ for any commutative ring $K$ [2511.03869]. Noncommutative Stone duality provides the topological content: étale Boolean right-restriction monoids are dual to étale groupoids with Stone spaces of identities. Under this duality, partial units correspond to compact-open bisections, and general elements to compact-open local right-bisections. Determining when a Boolean restriction semigroup admits a cosupport ($^+$) so as to be a Boolean range semigroup is characterized by openness of the range map in the germ category; uniqueness is established by algebraic axiomatization [2410.22181]. ## 6. Examples and Applications | Construction | Boolean Restriction Semigroup (BRS) Example | Companion Structures | |----------------------------------|--------------------------------------------|----------------------------| | Partial maps on set $X$ | $\mathrm{PT}(X)$ | $\mathsf{I}(X)$ (inverse) | | Clopen homeomorphisms of $X$ | $\mathsf{S}(X)$ (local homeos) | $\mathsf{I}^{cp}(X)$ | | Polycyclic/Cuntz/Thompson | $H_n=\mathsf{Etale}(C_n)$ | $C_n$, $G_{n,1}$ | Concrete operator-algebraic and dynamical applications arise via the study of the convolution algebras of the associated germ categories, extending the actions considered in groupoid C*-algebras and Steinberg algebras. The Boolean restriction structure permits the modeling of systems with partial symmetries that are not involutive, as in certain non-self-adjoint operator algebras. ## 7. Synthesis and Theoretical Significance A Boolean restriction semigroup $(S,\,^*,\cdot)$ is a natural one-sided generalization and completion of Boolean inverse monoids, with algebraic structure precisely reflecting the partial logic of locally defined transformations and Boolean logic on the projections. All such semigroups arise from — and are equivalent to — ample topological categories, providing a powerful algebraic-topological duality which subsumes and broadens the scope of inverse semigroup/groupoid duality. This framework organizes the algebraic logic of partial maps, symmetries, and Boolean data, and interfaces directly with the structure of zero-dimensional spaces, operator algebras, and the categorical geometry of groupoids [2511.03869][2404.08606][2410.22181]. Source: https://www.emergentmind.com/topics/boolean-restriction-semigroup