Geodesic
Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 3, pp. 63-103.

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In this work, lattice isomorphisms of semirings $C^+(X)$ of continuous nonnegative functions over an arbitrary topological space $X$ are characterized. It is proved that any isomorphism of lattices of all subalgebras with a unit of semirings $C^+(X)$ and $C^+(Y)$ is induced by a unique isomorphism of semirings. The same result is also correct for lattices of all subalgebras excepting the case of two-point Tychonovization of spaces. 
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E. M. Vechtomov; V. V. Sidorov. Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 3, pp. 63-103. http://geodesic.mathdoc.fr/item/FPM_2010_16_3_a2/

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