Unification types and union splittings in intermediate logics

We classify intermediate logics according to their unification types. There are exactly two minimal intermediate logics with hereditary finitary unification: the least logic with hereditary unitary unification and the least logic with hereditary projective proximity (a notion close to projective approximation of Ghilardi [17], [18]), see Figure 4. They are locally tabular and are union splittings in the lattice Ext INT. There are exactly four maximal intermediate logics with nullary unification (see Figure 21) and they are tabular. Any intermediate logic with neither hereditary unitary unification nor with hereditary projective proximity is included in one of the four logics. There are logics with finitary/unitary (but not hereditary finitary) unification scattered among the majority of those with nullary unification, see Figure 23. Our main tools are the characterization of locally tabular logics with finitary (or unitary) unification, by their Kripke models [12], [13] and splittings.