# Make own rule for multiplication

I'm new to mathematica, so I still cannot use it properly. I want to do symbolic programming.

My question is : is there any way to define our own multiplication. Suppose $a,b$ are arbitrary variables, I want to define $ba=qab$ as the rule, with $q$ is some constant. Then if a compute $baab$, then I want the result become $baab=qabab=q^{2}aabb=q^{2}a^{2}b^{2}$ (I swap the position of the first $b$ from left to $a$ two times).

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– user9660
Nov 20 '15 at 13:50
• You might consider using NonCommutativeMultiply[] in your implementation. Nov 20 '15 at 13:56
• I'll try to look at that command and how it works then Nov 20 '15 at 13:59
• yes, I want the general rules to be $ba=qab$, so the multiplication is not commutative. Nov 20 '15 at 18:18
• Can all the q's move all the way to the left in any expression? $-$ that is, do they commute with a and b? Nov 20 '15 at 18:49

As suggested in comment by J.M., NonCommutativeMultiply might be useful here. Using //. and two replacement rules you can get desired results.

$ncmRules = { (* Change b ** a to q a ** b. *) x___ ** b^n_. ** a^m_. ** y___ :> q^(n m) x ** a^m ** b^n ** y, (* Replace adjacent powers of same multiplicands by single power. *) x___ ** y_^n_. ** y_^m_. ** z___ :> x ** y^(n + m) ** z }; a ** b //.$ncmRules
(* a ** b *)
b ** a //. $ncmRules (* q a ** b *) b ** a ** a ** b //.$ncmRules
(* q^2 a^2 ** b^2 *)
b ** a ** b ** a ** a //. $ncmRules (* q^5 a^3 ** b^2 *) b ** a ** b ** b ** a //.$ncmRules
(* q^4 a^2 ** b^3 *)

• I haven't check this page for a week, many thanks @jkuczm , it works Nov 30 '15 at 17:42

Define your multiplication by two rules

CircleTimes[x_, y_] := q Times[x, y]


for 2 arguments and

CircleTimes[a___] := Module[{b, c},
If[Length[{a}] > 2,
b = CircleTimes [{a}[[1]], {a}[[2]] ];
c = Join[{b}, {a}[[3 ;; All]] ];
Apply[CircleTimes, c]]
]


It can be written shorter, here I separated into steps for clarity.

Test:

a⊗b⊗b⊗a
(*a^2 b^2 q^3*)

a⊗b a⊗b
(*a^2 b^2 q^2*)


An approach similar to that of yarchik but non-commutative is

CircleTimes[a, b] := Times[a, b]
CircleTimes[b, a] := q Times[a, b]
CircleTimes[z_, z_] := Times[z, z]
CircleTimes[z__] := Module[{zz = {z}, tem}, tem = CircleTimes @@ zz[[-2 ;; -1]];
(CircleTimes @@ Join[zz[[1 ;; -3]], {First@tem}]) Rest@tem]


Then,

a⊗b
(* a b *)

b⊗a
(* a b q *)

a⊗b⊗b⊗a
(* a^2 b^2 q^2 *)


as specified in the Question. Note that this definition of CircleTimes works only for a and b, because those are the only symbols defined in the Question. It could be generalized to other symbols if the OP wished to provide rules for them.