Mixed modal - sufficiency algebras and complex algebras of frames

A mixed modal - sufficiency (MS) algebra is a structure  where B 
is a Boolean algebra, f is a modal (i.e. normal and additive) operator 
and g is a sufficiency (i.e. co-normal and co-additive) operator.  The 
mixed canonical frame of such an algebra is the relational structure 
 where R, S are binary relations arising from f and g, 
satisfying certain canonical conditions. Conversely, a frame  
naturally leads to a Boolean algebra ,[[S]]> where B is the power 
set algebra of X,  is the possibility operator induced by R and [[S]] 
the sufficiency operator induced by S. Such a structure is called a 
mixed complex algebra of .

This seminar will report several new results on various classes of MS - 
algebras. I shall define the classes of mixed algebras (MIA), weak mixed 
algebras (WMIA), and K ~ mixed  algebras (KMIA). These classes are 
strictly contained in each other, and

1. MIA is not first order definable.
2. WMIA is a universal (but not equational) class.
3. KMIA is an equational class.

It turns out that KMIA is the equational class generated by WMIA, and 
also the equational class generated by the complex algebras of frames 
. Furthermore, KMIA is the class of algebras appropriate to the 
Logic K~ of Gargy, Passy and Tinchev (1987).

The results were obtained  jointly with Ewa Orlowska (ITL Warsaw) and 
Tinko Tinchev (Sofia University).