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Coherent spaces, Boolean rings and quantum gates

Coherent spaces spanned by a finite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean ring (Stone\rq{}s formalism). Applications of this Boolean ring into classical CNOT gates with $n$-ary variables, and also quantum CNOT gates with coherent states, are discussed.