* Institutions 
	formalization of spec. languages
		Reference Goguen-Burstall84 (LNCS 164)

**	vocabulary: category (Sign) of signatures and signature morphisms
		e.g., category of equational signatures
			objects: pairs (S,OPS)
				where S is set of sort names
				OPS is a family f operator names
					(indexed by signatures in S* x S)
			e.g., \Sigma=(S,OPS), where
			     S = {Bool, Int}
			     OPS_{(Bool,Bool),Bool} = {and, or, implies}

			morphisms: Phi: (S,OPS) -> (S',OPS')
				is a map f: S -> S'
				and g_{u,s}: OPS_{u,s} -> OPS_{f*(u),f(s)}
					a S* x S indexed family of maps
			    i.e., preserves sorts of operations
			    e.g., f(Bool) = Boolean
				  g_{(Bool,Bool),Bool}(and) = "&"

**	set of sentences over a signatures (functor Sen: Sign -> Set)
		a \Sigma-equation is a triple (X, t1, t2),
			where X is a S-indexed set (of variables)
			t1 and t2 are terms over X with the same sort
		e.g., Eqn: \Sigma -> set of $\Sigma-equations
		      Eqn: signature morphisms -> maps on equations

**	category of models of a signature (functor Mod: Sign -> Cat^{op})
		a variety, set of models with same signature

		Alg(\Sigma) = category of \Sigma-algebras
			A = (|A|,OPS^A)
			where (\Sigma=(S,OPS))
			|A| is carrier set = S-indexed family of sets (A_s)
			OPS^A is a S* x S indexed family of maps
				alpha: OPS_{u,s} -> A_u -> A_s
				if g in OPS_{u,s}, write g^A for alpha(g)
		homomorphisms (in the category Mod(\Sigma)):
			f: A->B, where both are \Sigma-algebras
				is S-indexed family of maps f_s:A_s -> B_s
			such that for all g: u -> s (g in OPS_{u,s})
				where u = s1,s2,...,sn
				f_s(g^A(a1,...an)) = g^B(f_s1(a1),...,f_sn(an))

		i.e., homomorphisms have a substitution property
		e.g., A_Bool = {true, false}
		      and^A(b1,b2) = b1 & b2

		functor Mod:
			object map: Mod(\Sigma) = Alg(\Sigma)
			arrow map:
				maps signature morphism
					Phi = (f,g): \Sigma -> \Sigma'
				to functor:
					Alg(Phi): Alg(\Sigma') -> Alg(\Sigma)
					objects: Alg(Phi)(A') = A
						where A = (|A|,OPS^A)
							A_s = A'_{Phi(s)}
							OPS^A = Phi;OPS^A'
					arrows: Alg(Phi)(h': A' -> B') = h
						where h_s = h'_{Phi(s)}

			e.g., Alg(Phi)(Boolean) = Bool
				Alg(Phi)(&^B) = and^A

***	satisfaction relation |= between Mod(\Sigma) and Sen(\Sigma)
		for each \Sigma in Sign
	such that for each Phi: \Sigma -> \Sigma' in Sign
		for each m' in Mod(\Sigma'), for each e in Sen(\Sigma)
			m' |= Sen(Phi)(e) iff Alg(Phi)(m') |= e.

		e.g., suppose e' is obtained by translating e, by Phi 
			e.g., e is true and true = true
			      e' is true & true = true
		      suppose Alg(Phi)(m) = m',
				so that m has sorts Bool, Int, ...
			m' |= e' iff m |= e

** theory = set of sentences closed under implication
***	presentation = pair of signature and set of sentences
***	satisfaction
		a model satisfies presentation if satisfies each sentence
		A models (\Sigma,E) iff for each e in E, A |= e,
			write A |= E
***	closure notions
	E* is set of all A such that A |= E
	if M is a set of models,
		M* is set of all sentences satisfied by each m in M
		M* is the theory of M
	closure of set of sentences is E**
		E is closed iff E = E**
***	a theory is a theory presentation that is closed
		(\Sigma,E) such that E is closed
***	the theory presented by (\Sigma,E) is the closure of E