Categorical abstract algebraic logic: The largest theory system included in a theory family

ly, ← T may be characterized as the largest theory system of I ≤-included in the theory family T . Proposition 2.2 Suppose that I = 〈Sign, SEN, C〉 is a π-institution and T a theory family of I. Then ← T is the largest theory system of I that is ≤-included in T . P r o o f. Suppose that T ′ is a theory system of I, such that T ′ ≤ T . Let Σ ∈ |Sign|, and φ ∈ SEN(Σ), such that φ ∈ T ′ Σ. Therefore, since T ′ is, by hypothesis, a theory system of I, for all Σ′ ∈ |Sign|, f ∈ Sign(Σ,Σ′), SEN(f)(φ) ∈ T ′ Σ′ . But, also by hypothesis, T ′ ≤ T , whence we obtain SEN(f)(φ) ∈ TΣ′ , for all Σ′ ∈ |Sign|, f ∈ Sign(Σ,Σ′). But this is equivalent to φ ∈ SEN(f)−1(TΣ′), for all Σ′ ∈ |Sign|, f ∈ Sign(Σ,Σ′), i. e., that φ ∈ {SEN(f)−1(TΣ′) : Σ′ ∈ |Sign|, f ∈ Sign(Σ,Σ′)} = ← TΣ. Therefore T ′ Σ ⊆ ← TΣ, for all Σ ∈ |Sign|, and, hence T ′ ≤ ← T . c © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org Math. Log. Quart. 52, No. 3 (2006) 291 It is now obvious that the following holds: Corollary 2.3 Suppose that I = 〈Sign, SEN, C〉 is a π-institution and T a theory family of I. T is a theory system of I if and only if ← T = T . P r o o f. The largest theory system that is included in a given theory family is the theory family itself if and only if the theory family is a theory system. Now use Proposition 2.2. Corollary 2.4 Suppose that I = 〈Sign, SEN, C〉 is a π-institution and T, T ′ theory families of I. If T ≤ T ′,