  Different versions of the notion of a state have been formulated for various
so-called quantum structures. In this paper, we investigate the interplay among
states on synaptic algebras and on its sub-structures. A synaptic algebra is a
generalization of the partially ordered Jordan algebra of all bounded
self-adjoint operators on a Hilbert space. The paper culminates with a
characterization of extremal states on a commutative generalized Hermitian
algebra, a special kind of synaptic algebra.
