Embedding theorems for classes of GBL-algebras

The poset product construction is used to derive embedding theorems for several classes of generalized basic logic algebras (GBL-algebras). In particular it is shown that every n-potent GBL-algebra is embedded in a poset product of finite n-potent MV-chains, and every normal GBL-algebra is embedded in a poset product of totally ordered GMV-algebras. Representable normal GBL-algebras have poset product embeddings where the poset is a root system. We also give a Conrad–Harvey–Holland-style embedding theorem for commutative GBL-algebras, where the poset factors are the real numbers extended with −∞. Finally, an explicit construction of a generic commutative GBL-algebra is given, and it is shown that every normal GBL-algebra embeds in the conucleus image of a GMV-algebra.