Abstract
We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what we term the ‘Booleanization’. We establish the exact theoretical connection between the Booleanization of an inverse semigroup and Paterson’s universal groupoid of the inverse semigroup and we explicitly compute the concrete Booleanization of the polycyclic inverse monoid \(P_{n}\) and demonstrate its affiliation with the Cuntz–Toeplitz algebra.
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Notes
One could argue that the term ‘compact’ is more appropriate, but we follow the terminology in [5].
Observe that this notation refers only to proper filters.
This set is also termed the ‘set of units’ in the literature and denoted by \(G^{(0)}\).
These are frequently referred to in the literature as ample groupoids. We prefer our term because it makes clear that we are generalizing classical Stone duality.
We have modified Paterson’s construction in the obvious way to deal with the case where the inverse semigroup has a zero.
Compare with [5].
A Boolean ring is a ring in which every element is an idempotent. A simple exercise shows that such rings are always commutative.
A key part of our computation is the description of the Stone space of the set of all reverse definite languages over a fixed alphabet. This space is actually described in [22] though for completely different reasons from ours.
We have borrowed this terminology from [4].
Strictly speaking, this should be reverse definite.
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The author is very grateful to the anonymous referee for their careful reading of this paper.
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Communicated by Benjamin Steinberg.
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Lawson, M.V. The Booleanization of an inverse semigroup. Semigroup Forum 100, 283–314 (2020). https://doi.org/10.1007/s00233-019-10071-8
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DOI: https://doi.org/10.1007/s00233-019-10071-8