This defines the unary operation
x [right arrow] [bar.x] on each epigroup; we call this operation pseudoinversion.
Let M be a ([logical not], A, V, [right arrow])-type algebra, where [logical not] is a unary operation
and [conjuncton], [disjunction], and [right arrow] are binary operations.
An MV-algebra M = (M; 0, [logical not], [direct sum]) is an algebra where [direct sum] is an associative and commutative binary operation on M having 0 as the neutral element, a unary operation
[logical not] is involutive with a [direct sum] [logical not]0 = [logical not]0 for all a [member of] M, and moreover the identity a [direct sum] [logical not](a [direct sum] [logical not]b) = b [direct sum] [logical not](b [direct sum] [logical not]a) is satisfied for all a,b [member of] M.
Intensional algebra for the intensional FOL in Definition 1 is a structure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with binary operations [conj.sub.s] : [D.sub.j] x[D.sub.I] [right arrow] [D.sub.i], unary operation
neg : [D.sub.I] [right arrow] [D.sub.I], unary operations
[exists.sub.n] : [D.sub.I] [right arrow] [D.sub.I], such that for any extensionalization function he W, and u [member of] [D.sub.k], [upsilon] [member of] [D.sub.j], k, j [greater than or equal to] 0,
* that is a unary operation
interpreted so that('[s.sub.A]') denotes the class of models determined by the [L.sub.A]-sentence [s.sub.A].
Moreover, the unary operation
[sup.-1] of <Q, [[direct sum].sub.k]> will be denoted by [[??].sub.k] and the element [[??].sub.k] x coincides with k/x.
Then the meaning of the rules of constructing well-formed formulas P,..., P can be explained as follows: for each k from 1 to 10, the rule P[k] determines a partial unary operation
Op[k] on the set Degr(B) with the value being an element of Ls(B).
An algebra (A, [conjunction], [disjunction],[(-).sup.~]) where A is non-empty set with 1, [conjunction], [disjunction] are binary operations and [(-).sup.-] is a unary operation
and a unary operation
*: S [right arrow] S satisfying the following three axioms: