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-**N**atural meta-number is redundant. [The principle of “additive idempotency”, or of “unquantifiability”].”

**( qn ) [ qn + qn = qn ]**.

**Rule 7**. The Addition of one “meta-**N**atural meta-number” to a different one * does not reduce* to any single “meta-

**N**atural 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 **q**ualifiers multiplies ontology [ M.D.: i.e., adds new ontology [represented by additional ontological **q**ualifiers, 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 **q**ualifiers”, as primitives of this axioms-system.

Rule **8** so defines “ontological multiplication”, or “qualitative multiplication” of “category meta-numbers”, or of “ontological-categorial **q**ualifiers”,

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**.