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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.10221 |
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| _version_ | 1866909291522293760 |
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| author | Biswas, Pronay Bag, Sagarmoy Sardar, Sujit Kumar |
| author_facet | Biswas, Pronay Bag, Sagarmoy Sardar, Sujit Kumar |
| contents | For a measurable space $(X,\mathcal{A})$, let $\mathcal{M}^+(X,\mathcal{A})$ be the commutative semiring of non-negative real-valued measurable functions with pointwise addition and pointwise multiplication. We show that there is a lattice isomorphism between the ideal lattice of $\mathcal{M}^+(X,\mathcal{A})$ and the ideal lattice of its ring of differences $\mathcal{M}(X,\mathcal{A})$.
Moreover, we infer that each ideal of $\mathcal{M}^+(X,\mathcal{A})$ is a semiring $z$-ideal.
We investigate the duality between cancellative congruences on $\mathcal{M}^{+}(X,\mathcal{A})$ and $Z_{\mathcal{A}}$-filters on $X$. We observe that for $σ$-algebras, compactness and pseudocompactness coincide, and we provide a new characterization for compact measurable spaces via algebraic properties of $\mathcal{M}^+(X,\mathcal{A})$. It is shown that the space of (real) maximal congruences on $\mathcal{M}^+(X,\mathcal{A})$ is homeomorphic to the space of (real) maximal ideals of the $\mathcal{M}(X,\mathcal{A})$. We solve the isomorphism problem for the semirings of the form $\mathcal{M}^+(X,\mathcal{A})$ for compact and realcompact measurable spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concerning semirings of measurable functions Biswas, Pronay Bag, Sagarmoy Sardar, Sujit Kumar Functional Analysis Rings and Algebras Primary 54C40, Secondary 46E30 For a measurable space $(X,\mathcal{A})$, let $\mathcal{M}^+(X,\mathcal{A})$ be the commutative semiring of non-negative real-valued measurable functions with pointwise addition and pointwise multiplication. We show that there is a lattice isomorphism between the ideal lattice of $\mathcal{M}^+(X,\mathcal{A})$ and the ideal lattice of its ring of differences $\mathcal{M}(X,\mathcal{A})$. Moreover, we infer that each ideal of $\mathcal{M}^+(X,\mathcal{A})$ is a semiring $z$-ideal. We investigate the duality between cancellative congruences on $\mathcal{M}^{+}(X,\mathcal{A})$ and $Z_{\mathcal{A}}$-filters on $X$. We observe that for $σ$-algebras, compactness and pseudocompactness coincide, and we provide a new characterization for compact measurable spaces via algebraic properties of $\mathcal{M}^+(X,\mathcal{A})$. It is shown that the space of (real) maximal congruences on $\mathcal{M}^+(X,\mathcal{A})$ is homeomorphic to the space of (real) maximal ideals of the $\mathcal{M}(X,\mathcal{A})$. We solve the isomorphism problem for the semirings of the form $\mathcal{M}^+(X,\mathcal{A})$ for compact and realcompact measurable spaces. |
| title | Concerning semirings of measurable functions |
| topic | Functional Analysis Rings and Algebras Primary 54C40, Secondary 46E30 |
| url | https://arxiv.org/abs/2408.10221 |