Strict Shells

The hypothesis that shells be Baer–Stone is rather restrictive. We weaken it, but to obtain a theorem comparable to that of the previous chapter, we must also weaken the previous conclusion: the representation is only as a subalgebra of the algebra of all global sections of a sheaf, rather than as an isomorphic image. The stalks still have no divisors of zero.The first section proves the theorem just summarized, for algebras that have only a multiplication and a nullity, called strict half-shells. We assume these half-shells have no nilpotents and satisfy null-symmetry, a collection of implications between products that are zero.The second section starts by exploring the consequence of having no divisors of zero when the sheaf is a Boolean product: the enveloping half-shell is then Baer–Stone. When the sheaf space is extremal disconnected, the enveloping half-shell is completely Baer–Stone.The third section adds a unity and an addition that is loop to the half-shell; this makes for stronger and simpler conclusions. We apply it to clusters.