A union is written:

SP₁ and ... and SP_n

When the current local environment is empty, each SP_i must determine a complete signature Sigma_i. The signature of the union is obtained by the ordinary union of the Sigma_i (not their disjoint union). Thus all (non-localized) occurrences of a symbol in the SP_i are interpreted uniformly (rather than being regarded as homonyms for potentially different entities).

The models are those models of the union signature for which the reduct along the signature inclusion morphism from SP_i is a model of SP_i, for each i=1,...,n.

When the current local environment is non-empty, each SP_i must determine an extension from it to a complete signature Sigma_i; then the resulting signature is determined as above. Similarly, models of the local environment are extended to models of the SP_i; then the resulting models are determined as above. This provides the required partial functions from signatures to signatures, and from model classes to model classes.