CHAPTER 2

Definition

Throughout this article, we will use the following notation. If S is a group,

then we define 5* := S\ {neutral element}. If S is a set which contains an element

called "0", then we define S* := S \ {0}. It will always be clear from the context

which definition we mean.

Consider an abelian group (V,+) and a (possibly non-abelian) group (W, EB).

The inverse of an element w G W will be denoted by 3w, and by

WIBHJ

2

,

we mean

wi E B (E\w2). Suppose that there is a map ry from V x W to V and a map TW from

W x V to W, both of which will be denoted by • or simply by juxtaposition, i.e.

rv(v,w) — vw — v - w and rw(w,v) = wv = w • v for all v G V and all w G W.

Consider a map F from V x V to W and a map i l from W x W to V, both of

which are "bi-additive" in the sense that

F(v! + v2lv) = F(Vl,v) E B F(v2,v);

F(v, vi + v2) = F(u, vi) ffl F(v, v2);

H(wi E B 1^2, w) = H(wi, w) + H(w2, w);

if (io, t^i E B ^2) = #(w, wi) + if (w;, ^2);

for all v,vi,v2 G V and all w,wi,w2 G W. Suppose furthermore that there exists

a fixed element e G V* and a fixed element ( 5 G W*, and suppose that, for each

v G V*, there exists an element v

_ 1

G V*, and for each w G VK*, there exists an

element K(W) G W*, such that, for all w,wi,w2 G VF and all v,vi,v2 G V, the

following axioms are satisfied. We define

Rad(F)

Rad(#)

Im(F)

Im(ff)

= eF(e,v) — v

= {veV I F(v,y) = 0}

= {w E ^ I #(w,W0 =0}

= F(V,V)

= H{W1W)

(Qx) we = w.

(Q2) ^ = ^.

(Q3) (^1 ffl ^2)^ = ^ 1 ^ ffl U^-

(Q4) (^1 +v2)w = viw + v2w.

(Q5) M~

e

) • ^ = ^ ( - v ) .

(Qe) v • w(—e) — vw.

(Q7) Im(F) C Rad(if).

(Q8) [WI,W2V]E = F(H(W2,WI),V).

(Q9) ( 5 G Rad(fi).

(Q10) If Rad(F) / 0, then e G Rad(F).

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