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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.11186 |
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| _version_ | 1866913942665691136 |
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| author | Sokolova, Ana Woracek, Harald |
| author_facet | Sokolova, Ana Woracek, Harald |
| contents | Convex semilattices are algebras that are at the same time a convex algebra and a semilattice, together with a distributivity axiom. These algebras have attracted some attention in the last years as suitable algebras for probability and nondeterminism, in particular by being the Eilenberg-Moore algebras of the nonempty finitely-generated convex subsets of the distributions monad.
A convex semilattice is cancellative if the underlying convex algebra is cancellative. Cancellative convex algebras have been characterized by M. H. Stone and by H. Kneser: A convex algebra is cancellative if and only if it is isomorphic to a convex subset of a vector space (with canonical convex algebra operations).
We prove an analogous theorem for convex semilattices: A convex semilattice is cancellative if and only if it is isomorphic to a convex subset of a Riesz space, i.e., a lattice-ordered vector space (with canonical convex semilattice operations). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cancellative Convex Semilattices Sokolova, Ana Woracek, Harald Logic in Computer Science Convex semilattices are algebras that are at the same time a convex algebra and a semilattice, together with a distributivity axiom. These algebras have attracted some attention in the last years as suitable algebras for probability and nondeterminism, in particular by being the Eilenberg-Moore algebras of the nonempty finitely-generated convex subsets of the distributions monad. A convex semilattice is cancellative if the underlying convex algebra is cancellative. Cancellative convex algebras have been characterized by M. H. Stone and by H. Kneser: A convex algebra is cancellative if and only if it is isomorphic to a convex subset of a vector space (with canonical convex algebra operations). We prove an analogous theorem for convex semilattices: A convex semilattice is cancellative if and only if it is isomorphic to a convex subset of a Riesz space, i.e., a lattice-ordered vector space (with canonical convex semilattice operations). |
| title | Cancellative Convex Semilattices |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2507.11186 |