Eight of the core rules of the F.E.D. “rules-system”, or “axioms-system”, for their “First Psychohistorical Algebra”, which answer your questions, were given in a letter dd. 09 June 2009, by Aoristos Dyosphainthos –
http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf
– and a fuller list, by Hermes de Nemores, in a “Preface” to an essay by “J2Y”, dated 20 February of this year [see p. 2] –
http://www.dialectics.org/dialectics/Briefs_files/Hermes_de_Nemores,F.E.D._Preface_to_New_Guest_Author_E.D._Brief_5,revision,posted_20FEB2013.pdf
However, it is the last three “rules of purely-qualitative calculation” that address your questions most directly –
Rule 6. "Self-Addition [or “doubling”] of any “meta-Natural meta-number is redundant. [The principle of “additive idempotency”, or of “unquantifiability”].”
( qn ) [ qn + qn = qn ].
Rule 7. The Addition of one “meta-Natural meta-number” to a different one does not reduce to any single “meta-Natural meta-number” value.
( qj )( qk ) [ [ qj not equal to qk ] ==> [ there is no qx such that qj + qk = qx ] ].
Rule 8. [Mutual] Multiplication of ontological qualifiers multiplies ontology [ M.D.: i.e., adds new ontology [represented by additional ontological qualifiers, with higher subscript-index than those extant prior to the performance of the multiplication operation] ].
( qj )( qk )[ qj x qk = qk + qj+k ].
Rules 6 and 7 define “ontological addition”, or “qualitative addition” of “category meta-numbers”, or of “ontological-categorial qualifiers”, as primitives of this axioms-system.
Rule 8 so defines “ontological multiplication”, or “qualitative multiplication” of “category meta-numbers”, or of “ontological-categorial qualifiers”,
F.E.D. usually uses a “plus sign” with a box around it for “qualitative addition”, and ususally uses an “x” sign with a box around it for “multiplication of qualities” [of “qualifiers”], to distinguish each operation from the “+” and the “x” operations of ordinary, “purely quantitative arithmetics”.
For the example you cited, if we identify / assign / interpret X or qX to be represented by q1,
which F.E.D. signs by –
q1 [—> X = qX
– then, indeed –
X**^2** = X x X = X + DX = X + q****XX = X + Y
– just as –
q1^2 = q1 x q1 = q1 + q1+1 = q1 + q2
– and –
DX ** <—] Dq1 = q2**
– and –
Y <—] q2.