Geodesic
Semirings of continuous partial numerical functions with extended addition
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 1, pp. 56-66 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article deals with the semiring of all continuous functions on a topological space $X$ with values in the topological field of real numbers $\mathbb{R}\cup\{\varnothing\}$, which is completed by the isolated zero $\varnothing$. Operations of addition and multiplication over functions are pointwise. This semiring coincides with the semiring $CP(X)$ of all continuous partial real-valued functions whose domains are clopen subsets of the topological space $X$. The maximal ideals and maximal congruences of the semirings $CP(X)$ are described. A class of maximal subalgebras in the semirings $CP(X)$ is found. It is proved that any Hewitt space $X$ is defined by the semiring $CP(X)$. The case of a finite discrete space $X$ is studied.
Keywords: extended field of real numbers, topological space, semiring of continuous functions, partial function, ideal, congruence, definability.
Mots-clés : subalgebra 
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E.M. Vechtomov; E. N. Lubyagina. Semirings of continuous partial numerical functions with extended addition. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 1, pp. 56-66. http://geodesic.mathdoc.fr/item/TIMM_2023_29_1_a3/

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