Abstract
We prove that for X Є ǀEǀ, E elementary topos, the following properties are equivalent: (a) X is K-finite, (b) for every upper semilattice B, the diagonal B → BX has a left adjoint.
References
Acuña–Ortega, O. (1977) Finiteness in Topoi. Ph. D. Dissertation, Wesleyan University, Middletown, CT.
Acuña–Ortega, O. (2011) “Una nota sobre objetos k-finitos en un topos Booleano con el objeto de los números naturales”, Revista de Matemática: Teoría y Aplicaciones 19(2): 239–245.
Birkhoff, G. (1973) Lattice Theory, 3rd Ed. American Mathematical Society Colloquium Publications, Volume XXV, Providence RI.
Nachbin, L. (1949) “On a characterization of the lattice of all ideals of a Boolean ring”, Fundamenta Mathematicae, 36(1): 137–142.
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