# Cohomology Theory of Nonassociative Algebras with Metagroup Relations

## Abstract

**:**

## 1. Introduction

## 2. Cohomology Theory of Nonassociative Algebras

**Definition**

**1.**

- (1)
- For each a and b in G, there is a unique $x\in G$ with $ax=b$ and
- (2)
- A unique $y\in G$ exists, satisfying $ya=b$, which is denoted by $x=a\backslash b=Di{v}_{l}(a,b)$ and $y=b/a=Di{v}_{r}(a,b)$ correspondingly,
- (3)
- A neutral (i.e., unit) element ${e}_{G}=e\in G$ exists: $\phantom{\rule{3.33333pt}{0ex}}eg=ge=g$ for each $g\in G$.The set of all elements $h\in G$ commuting and associating with G is
- (4)
- $Com(G):=\{a\in G:\forall b\in G,\phantom{\rule{3.33333pt}{0ex}}ab=ba\}$,
- (5)
- ${N}_{l}(G):=\{a\in G:\forall b\in G,\forall c\in G,\phantom{\rule{3.33333pt}{0ex}}(ab)c=a(bc)\}$,
- (6)
- ${N}_{m}(G):=\{a\in G:\forall b\in G,\forall c\in G,\phantom{\rule{3.33333pt}{0ex}}(ba)c=b(ac)\}$,
- (7)
- ${N}_{r}(G):=\{a\in G:\forall b\in G,\forall c\in G,\phantom{\rule{3.33333pt}{0ex}}(bc)a=b(ca)\}$,
- (8)
- $N(G):={N}_{l}(G)\cap {N}_{m}(G)\cap {N}_{r}(G)$;$\mathcal{C}(G):=Com(G)\cap N(G)$ is called the center $\mathcal{C}(G)$ of G.We call G a metagroup if a set G possesses a single-valued binary operation and satisfies Conditions (1)–(3) and
- (9)
- $(ab)c={\mathsf{t}}_{3}(a,b,c)a(bc)$for each a, b, and c in G, where ${\mathsf{t}}_{3}(a,b,c)\in \mathbf{\Psi}$, $\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}\subset \mathcal{C}(G)$;where ${\mathsf{t}}_{3}$ shortens a notation ${\mathsf{t}}_{3,G}$, where Ψ denotes a (proper or improper) subgroup of $\mathcal{C}(G)$.Then G will be called a central metagroup if, in addition to $(9)$, it satisfies the condition
- (10)
- $ab={\mathsf{t}}_{2}(a,b)ba$for each a and b in G, where ${\mathsf{t}}_{2}(a,b)\in \mathbf{\Psi}$.

**Lemma**

**1.**

- (1)
- ${\{{a}_{1},\dots ,{a}_{n}\}}_{q(n)}={t}_{n}{\{{a}_{v(1)},\dots ,{a}_{v(n)}\}}_{u(n)}$.

**Proof.**

**Lemma**

**2.**

- (1)
- $\phantom{\rule{3.33333pt}{0ex}}b\backslash e=(e/b){\mathsf{t}}_{3}(e/b,b,b\backslash e)$;
- (2)
- $(a\backslash e)b=(a\backslash b){\mathsf{t}}_{3}(e/a,a,a\backslash e)/{\mathsf{t}}_{3}(e/a,a,a\backslash b)$;
- (3)
- $b(e/a)=(b/a){\mathsf{t}}_{3}(b/a,a,a\backslash e)/{\mathsf{t}}_{3}(e/a,a,a\backslash e)$.

**Proof.**

- (4)
- $b(b\backslash a)=a$, $\phantom{\rule{3.33333pt}{0ex}}b\backslash (ba)=a$;
- (5)
- $(a/b)b=a$, $\phantom{\rule{3.33333pt}{0ex}}(ab)/b=a$

**Definition**

**2.**

- (1)
- $sa=as$ for each s in $\mathcal{T}$ and a in G,
- (2)
- $s(ra)=(sr)a$ for each s and r in $\mathcal{T}$, and $a\in G$,
- (3)
- $r(ab)=(ra)b$, $\phantom{\rule{3.33333pt}{0ex}}(ar)b=a(rb)$, $\phantom{\rule{3.33333pt}{0ex}}(ab)r=a(br)$ for each a and b in G, $r\in \mathcal{T}$.

**Note**

**1.**

**Example**

**1.**

- (1)
- $a\overline{a}=N(a)1$ with $N(a)\in F$,
- (2)
- $a+\overline{a}=T(a)1$ with $T(a)\in F$,
- (3)
- $T(ab)=T(ba)$

- (4)
- $C(A,f)=A\oplus Al$,
- (5)
- $(a+bl)(c+dl)=(ac-f\overline{d}b)+(da+b\overline{c})l$ and
- (6)
- $\overline{(a+bl)}=\overline{a}-bl$

**Definition**

**3.**

- (1)
- $h{M}_{g}={M}_{hg}$ and ${M}_{g}h={M}_{gh}$,
- (2)
- $(bh){x}_{g}=b(h{x}_{g})$ and ${x}_{g}(bh)=({x}_{g}h)b$ and $b{x}_{g}={x}_{g}b$,
- (3)
- $(hs){x}_{g}={\mathsf{t}}_{3}(h,s,g)h(s{x}_{g})$ and $(h{x}_{g})s={\mathsf{t}}_{3}(h,g,s)h({x}_{g}s)$ and $({x}_{g}h)s={\mathsf{t}}_{3}(g,h,s){x}_{g}(hs)$

**Example**

**2.**

- (1)
- $\forall p\in \mathcal{T}[\mathsf{C}(G)]$, $\phantom{\rule{3.33333pt}{0ex}}p\xb7({x}_{0},\dots ,{x}_{n+1})=((p{x}_{0}),\dots ,{x}_{n+1})$ and$({x}_{0},\dots ,({x}_{n+1}p))=({x}_{0},\dots ,{x}_{n+1})\xb7p$ and$\forall j\in \{1,\dots ,n\}$, $\phantom{\rule{3.33333pt}{0ex}}p\xb7({x}_{0},\dots ,{x}_{n+1})=({x}_{0},\dots ,(p{x}_{j}),\dots ,{x}_{n+1})$ and$({x}_{0},\dots ,({x}_{j}p),\dots ,{x}_{n+1})=({x}_{0},\dots ,{x}_{n+1})\xb7p$,where $0\xb7({x}_{1},\dots ,{x}_{n})=0$;
- (2)
- $(xy)\xb7({x}_{0},\dots ,{x}_{n+1})={\mathsf{t}}_{3}\xb7(x\xb7(y\xb7({x}_{0},\dots ,{x}_{n+1})))$with ${\mathsf{t}}_{3}={\mathsf{t}}_{3}(x,y,b)$, (see also Formula $(9)$ in Definition 1 above);
- (3)
- ${\mathsf{t}}_{3}\xb7(({x}_{0},\dots ,{x}_{n+1})\xb7(xy))=(({x}_{0},\dots ,{x}_{n+1})\xb7x)\xb7y$ with ${\mathsf{t}}_{3}={\mathsf{t}}_{3}(b,x,y)$;
- (4)
- $(x\xb7({x}_{0},\dots ,{x}_{n+1}))\xb7y={\mathsf{t}}_{3}\xb7(x\xb7(({x}_{0},\dots ,{x}_{n+1})\xb7y))$ with ${\mathsf{t}}_{3}={\mathsf{t}}_{3}(x,b,y)$;
- (5)
- $x\xb7({x}_{0},\dots ,{x}_{n+1})={t}_{n+3}(x,{x}_{0},\dots ,{x}_{n+1};{v}_{0}(n+3);l(n+3))\xb7((x{x}_{0}),{x}_{1},\dots ,{x}_{n+1})$where ${\{x,{x}_{0},\dots ,{x}_{n+1}\}}_{{v}_{0}(n+3)}=x{\{{x}_{0},\dots ,{x}_{n+1}\}}_{l(n+2)}$,${\{{x}_{0},\dots ,{x}_{n+1}\}}_{l(n+2)}={\{{x}_{0},\dots ,{x}_{n}\}}_{l(n+1)}{x}_{n+1}$,${\{{x}_{0}\}}_{l(1)}={x}_{0}$, ${\{{x}_{0}{x}_{1}\}}_{l(2)}={x}_{0}{x}_{1}$;where $b={\{{x}_{0},\dots ,{x}_{n+1}\}}_{l(n+2)}$,${t}_{n}({x}_{1},\dots ,{x}_{n};u(n),w(n)):={t}_{n}({x}_{1},\dots ,{x}_{n};u(n),w(n)|id)$using the shortened notation;
- (6)
- $({x}_{0},\dots ,{x}_{n+1})\xb7x={t}_{n+3}({x}_{0},\dots ,{x}_{n+1},x;l(n+3),{v}_{n+2}(n+3))\xb7({x}_{0},\dots ,{x}_{n},({x}_{n+1}x))$for every $x,y,{x}_{0},\dots ,{x}_{n+1}$ in G, where $({x}_{0},\dots ,{x}_{n+1})$ denotes a basic element of ${K}_{n}$ over $\mathcal{T}$, corresponding to the left ordered tensor product$(\dots (({x}_{0}\otimes {x}_{1})\otimes {x}_{2})\dots \otimes {x}_{n})\otimes {x}_{n+1}$,${\{{x}_{0},\dots ,{x}_{n+1},x\}}_{{v}_{n+2}(n+3)}={\{{x}_{0},\dots ,{x}_{n},{x}_{n+1}x\}}_{l(n+2)}$.

**Proposition**

**1.**

**Proof.**

- (1)
- ${\partial}_{n}((x\xb7({x}_{0},{x}_{1},\dots ,{x}_{n},{x}_{n+1}))\xb7y)=$${\sum}_{j=0}^{n}{(-1)}^{j}\xb7{t}_{n+4}(x,{x}_{0},\dots ,{x}_{n+1},y;l(n+4),{u}_{j+1}(n+4))$$\xb7((x\xb7(<{x}_{0},{x}_{1},\dots ,{x}_{n+1}{>}_{j+1,n+2}))\xb7y)$, where
- (2)
- $<{x}_{0},\dots ,{x}_{n+1}{>}_{1,n+2}:=(({x}_{0}{x}_{1}),{x}_{2},\dots ,{x}_{n+1})$,
- (3)
- $<{x}_{0},\dots ,{x}_{n+1}{>}_{2,n+2}:=({x}_{0},({x}_{1}{x}_{2}),{x}_{3},\dots ,{x}_{n+1})$,...,
- (4)
- $<{x}_{0},\dots ,{x}_{n+1}{>}_{n+1,n+2}:=({x}_{0},\dots ,{x}_{n-1},({x}_{n}{x}_{n+1}))$,
- (5)
- ${\partial}_{0}(x\xb7({x}_{0},{x}_{1}))\xb7y=(x\xb7({x}_{0}{x}_{1}))\xb7y$,
- (6)
- ${\{{x}_{0},{x}_{1},\dots ,{x}_{n+1}\}}_{l(n+2)}:=(\dots (({x}_{0}{x}_{1}){x}_{2})\dots ){x}_{n+1}$;
- (7)
- ${\{x,{x}_{0},\dots ,{x}_{n+1},y\}}_{{u}_{1}(n+4)}:=(x{\{({x}_{0}{x}_{1}),{x}_{2},\dots ,{x}_{n+1}\}}_{l(n+1)})y$,...,
- (8)
- ${\{x,{x}_{0},\dots ,{x}_{n+1},y\}}_{{u}_{n+1}(n+4)}:=(x{\{{x}_{0},{x}_{1},\dots ,({x}_{n}{x}_{n+1})\}}_{l(n+1)})y$

- (9)
- ${\{{x}_{0},\dots ,{x}_{n+1}\}}_{{v}_{1}(n+2)}:={\{({x}_{0}{x}_{1}),{x}_{2},\dots ,{x}_{n+1}\}}_{l(n+1)}$,...,
- (10)
- ${\{{x}_{0},\dots ,{x}_{n+1}\}}_{{v}_{n+1}(n+2)}:={\{{x}_{0},{x}_{1},\dots ,({x}_{n}{x}_{n+1})\}}_{l(n+1)}$

- (11)
- ${\mathbf{s}}_{n}({x}_{0},\dots ,{x}_{n+1})=(1,{x}_{0},\dots .,{x}_{n+1})$ for every ${x}_{0},\dots ,{x}_{n+1}$ in G. From Formulas $(9)$ and $(10)$ in Definition 1 and $(1)$ in Lemma 1 the identities
- (12)
- ${t}_{n}({x}_{1},\dots ,{x}_{n};q(n),u(n)|v){t}_{n}({x}_{1},\dots ,{x}_{n};u(n),q(n)|{v}^{-1})=1$
- (13)
- ${t}_{n}({x}_{1},\dots ,{x}_{n};q(n),u(n)){t}_{n}({x}_{1},\dots ,{x}_{n};u(n),w(n))$$={t}_{n}({x}_{1},\dots ,{x}_{n};q(n),w(n))$ follow for every element ${x}_{1}$,...,${x}_{n}$ in metagroup G. Vectors $q(n)$, $u(n)$, and $w(n)$ indicate the orders of their multiplication, $v\in {S}_{n}$ and $n\in \mathbf{N}$. The following identity is evident:
- (14)
- ${t}_{n+1}(1,{x}_{1},\dots ,{x}_{n};q(n+1),u(n+1)|v(n+1))$$={t}_{n}({x}_{1},\dots ,{x}_{n};q(n),u(n)|v(n))$ for data $q(n)$, $u(n)$ and $v(n)$ obtained from $q(n+1)$, $u(n+1)$, and $v(n+1)$ correspondingly by taking the identity $1b=b1=b$ into account for each $b\in G$. Hence, ${\mathbf{s}}_{n}(({x}_{0},\dots ,{x}_{n+1})\xb7y)=({\mathbf{s}}_{n}({x}_{0},\dots ,{x}_{n+1}))\xb7y$ for every ${x}_{0},\dots ,{x}_{n+1},y$ in G.Let ${p}_{n}:{K}_{n+1}\to {K}_{n}$ be a $\mathcal{T}$-linear mapping, such that
- (15)
- ${p}_{n}(a\otimes b)=a\xb7b$ and ${p}_{n}(b\otimes a)=b\xb7a$ for each $a\in {K}_{n}$ and $b\in A$. Therefore, from Formulas $(13)$ and $(14)$, we deduce that ${p}_{n}{\mathbf{s}}_{n}=id$ is the identity on ${K}_{n}$. Consequently, ${\mathbf{s}}_{n}$ is a monomorphism.

- (16)
- ${\partial}_{n+1}{\mathbf{s}}_{n}+{\mathbf{s}}_{n-1}{\partial}_{n}=1$ for each $n\ge 0$ are fulfilled, where 1 denotes the identity operator on ${K}_{n}$. Therefore, the recurrence relation
- (17)
- ${\partial}_{n}{\partial}_{n+1}{\mathbf{s}}_{n}={\mathbf{s}}_{n-2}{\partial}_{n-1}{\partial}_{n}$ is accomplished, since ${\partial}_{n}{\partial}_{n+1}{\mathbf{s}}_{n}={\partial}_{n}(1-{\mathbf{s}}_{n-1}{\partial}_{n})={\partial}_{n}-({\partial}_{n}{\mathbf{s}}_{n-1}){\partial}_{n}={\partial}_{n}-(1-{\mathbf{s}}_{n-2}{\partial}_{n-1}){\partial}_{n}$.

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Notation**

**1.**

**Theorem**

**1.**

- (1)
- $0\to Ho{m}_{\mathcal{T}}{({L}_{0},M)}_{\overrightarrow{{\u03f5}^{*}}}Ho{m}_{\mathcal{T}}{({L}_{1},M)}_{\overrightarrow{{\delta}^{1}}}Ho{m}_{\mathcal{T}}{({L}_{2},M)}_{\overrightarrow{{\delta}^{2}}}$$Ho{m}_{\mathcal{T}}{({L}_{3},M)}_{\overrightarrow{{\delta}^{3}}}Ho{m}_{\mathcal{T}}{({L}_{4},M)}_{\overrightarrow{{\delta}^{4}}}\dots $

**Proof.**

- (1)
- $({x}_{0},\dots ,{x}_{n+1})={t}_{n+2}({x}_{0},\dots ,{x}_{n+1};l(n+2),w(n+2))\xb7$$(({x}_{0}\otimes ({x}_{1},\dots ,{x}_{n}))\otimes {x}_{n+1})$ and
- (2)
- $({x}_{0},\dots ,{x}_{n+1})={t}_{n+2}({x}_{0},\dots ,{x}_{n+1};l(n+2),w(n+2))\xb7$$(z\otimes ({x}_{1},\dots ,{x}_{n}))$, where $({x}_{1},\dots ,{x}_{n})$ is a basic element in ${L}_{n}$ for every ${x}_{0},\dots ,{x}_{n+1}$ in G, ${\{{x}_{0},\dots ,{x}_{n+1}\}}_{w(n+2)}=({x}_{0}{\{{x}_{1},\dots ,{x}_{n}\}}_{l(n)}){x}_{n+1}$, $z\in {A}^{e}$, $z={x}_{0}\otimes {x}_{n+1}^{*}$.

- (2)
- $f({x}_{1},\dots ,{x}_{n})={\sum}_{g\in G}{f}_{g}({x}_{1},\dots ,{x}_{n})$,

- (3)
- $(xy)\xb7{f}_{g}({x}_{1},\dots ,{x}_{n})={\mathsf{t}}_{3}(x,y,g)\xb7(x\xb7(y\xb7{f}_{g}({x}_{1},\dots ,{x}_{n})))$,
- (4)
- ${\mathsf{t}}_{3}(g,x,y)\xb7({f}_{g}({x}_{1},\dots ,{x}_{n})\xb7(xy))=({f}_{g}({x}_{1},\dots ,{x}_{n})\xb7x)\xb7y$,
- (5)
- $(x\xb7{f}_{g}({x}_{1},\dots ,{x}_{n}))\xb7y={\mathsf{t}}_{3}(x,g,y)\xb7(x\xb7({f}_{g}({x}_{1},\dots ,{x}_{n})\xb7y))$ for every g and $x,y,{x}_{1},\dots ,{x}_{n}$ in G, where coefficients ${\mathsf{t}}_{3}$ are prescribed by Formula $(9)$ in Definition 1. Also,
- (6)
- $x\xb7{f}_{g}({x}_{1},\dots ,{x}_{n}):=x\xb7({f}_{g}({x}_{1},\dots ,{x}_{n}))$ and
- (7)
- ${f}_{g}({x}_{1},\dots ,{x}_{n})\xb7y:=({f}_{g}({x}_{1},\dots ,{x}_{n}))\xb7y$.For $n=0$ and $g=e$, naturally, the identities are fulfilled:
- (8)
- $(xy)\xb7{f}_{e}(\phantom{\rule{3.33333pt}{0ex}})=x\xb7(y\xb7{f}_{e}(\phantom{\rule{3.33333pt}{0ex}}))$, $({f}_{e}(\phantom{\rule{3.33333pt}{0ex}})\xb7x)\xb7y={f}_{e}(\phantom{\rule{3.33333pt}{0ex}})\xb7(xy)$ and $(x\xb7{f}_{e}(\phantom{\rule{3.33333pt}{0ex}}))\xb7y=x\xb7({f}_{e}(\phantom{\rule{3.33333pt}{0ex}})\xb7y)$.

- (9)
- $({\delta}^{n}f)({x}_{1},\dots ,{x}_{n+1})=$${\sum}_{j=0}^{n+1}{(-1)}^{j}{t}_{n+1}({x}_{1},\dots ,{x}_{n+1};l(n+1),{u}_{j+1}(n+1))\xb7{[f,{x}_{1},{x}_{2},\dots ,{x}_{n+1}]}_{j+1,n+1}$, where
- (10)
- ${[f,{x}_{1},\dots ,{x}_{n+1}]}_{1,n+1}:={x}_{1}\xb7f({x}_{2},\dots ,{x}_{n+1})$, ${\{{x}_{1},\dots ,{x}_{n+1}\}}_{{u}_{1}(n+1)}={x}_{1}{\{{x}_{2},\dots ,{x}_{n+1}\}}_{l(n)}$;
- (11)
- ${[f,{x}_{1},\dots ,{x}_{n+1}]}_{2,n+1}:=f(({x}_{1}{x}_{2}),\dots ,{x}_{n+1})$, ${\{{x}_{1},\dots ,{x}_{n+1}\}}_{{u}_{2}(n+1)}={\{({x}_{1}{x}_{2}),\dots ,{x}_{n+1}\}}_{l(n)}$;...;
- (12)
- ${[f,{x}_{1},\dots ,{x}_{n+1}]}_{n+1,n+1}:=f({x}_{1},{x}_{2},\dots ,({x}_{n}{x}_{n+1}))$; ${\{{x}_{1},\dots ,{x}_{n+1}\}}_{{u}_{n+1}(n+1)}={\{{x}_{1},\dots ,({x}_{n}{x}_{n+1})\}}_{l(n)}$;
- (13)
- ${[f,{x}_{1},\dots ,{x}_{n+1}]}_{n+2,n+1}:=f({x}_{1},{x}_{2},\dots ,{x}_{n})\xb7{x}_{n+1}$, ${\{{x}_{1},\dots ,{x}_{n+1}\}}_{{u}_{n+2}(n+1)}={\{{x}_{1},\dots ,{x}_{n+1}\}}_{l(n+1)}=(\dots (({x}_{1}{x}_{2}){x}_{3})\dots {x}_{n}){x}_{n+1}$; with ${u}_{0}(n+1)=l(n+1)$.

- (14)
- For each $b\in G$, ${h}_{1,b}$ exists, so that ${h}_{1,b}:{K}_{n+1}\to {M}_{1}$ and ${f}_{b}={h}_{1,b}{L}_{b}$, where ${L}_{b}$ is the left multiplication operator on b:
- (15)
- $({h}_{1,b}{L}_{b})({x}_{1},\dots ,{x}_{n})=b\xb7({h}_{1,b}({x}_{1},\dots ,{x}_{n}))$ for every ${x}_{1},\dots ,{x}_{n}$ in G. Moreover, $zg=0$ (or $gz=0$) in $\mathbf{Z}[G]$ for $g\in G$ and $z\in \mathbf{Z}[G]$, if and only if $z=0$, since G is a metagroup.

- (16)
- $({\delta}^{0}f)(x)=xf(\phantom{\rule{3.33333pt}{0ex}})-f(\phantom{\rule{3.33333pt}{0ex}})x=0$ for each $x\in G$.

- (17)
- ${t}_{2}(x,y;l(2),{u}_{1}(2))\xb7x\xb7f(y)-$${t}_{2}(x,y;l(2),{u}_{2}(2))\xb7f(xy)+{t}_{2}(x,y;l(2),{u}_{3}(2))\xb7f(x)\xb7y$$=x\xb7f(y)-f(xy)+f(x)\xb7y=0$

- (18)
- $f(xy)=x\xb7f(y)+f(x)\xb7y$.

**Remark**

**1.**

- (1)
- ${\mathsf{t}}_{3}({x}_{1},{x}_{2},{x}_{3})\xb7{x}_{1}\xb7f({x}_{2},{x}_{3})+{\mathsf{t}}_{3}({x}_{1},{x}_{2},{x}_{3})\xb7f({x}_{1},({x}_{2}{x}_{3}))$$=f(({x}_{1}{x}_{2}),{x}_{3})+f({x}_{1},{x}_{2})\xb7{x}_{3}$

- (2)
- $f(x,y)=(\delta h)(x,y)={\sum}_{j=0}^{2}{(-1)}^{j}{t}_{2}(x,y;l(2),{u}_{j+1}(2))\xb7{[h,{x}_{1},{x}_{2}]}_{j+1,2}$,$=x\xb7h(y)-h(xy)+h(x)\xb7y$.

- (3)
- $0\to {M}_{\overrightarrow{\xi}}{P}_{\overrightarrow{\eta}}N\to 0$,

- (4)
- $0\to {M}_{\overrightarrow{{\xi}^{\prime}}}{{P}^{\prime}}_{\overrightarrow{{\eta}^{\prime}}}N\to 0$

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Definition**

**4.**

**Theorem**

**4.**

**Proof.**

- (1)
- $({x}_{1},\dots ,{x}_{n+1})\xb7y:={t}_{n+2}({x}_{1},\dots ,{x}_{n+1},y;l(n+2),{u}_{n+2}(n+2))\xb7({x}_{1},\dots ,{x}_{n},({x}_{n+1}y))$ and
- (2)
- $y\xb7({x}_{1},\dots ,{x}_{n+1})={\sum}_{j=1}^{n+1}{(-1)}^{j+1}\xb7$${t}_{n+2}(y,{x}_{1},\dots ,{x}_{n+1};{u}_{1}(n+2),{u}_{j+1}(n+2))\xb7<y,{x}_{1},{x}_{2},\dots ,{x}_{n+1}{>}_{j,n+2}$

- (3)
- $({x}_{0}\xb7f)({x}_{1},\dots ,{x}_{n})={x}_{0}\xb7(f({x}_{1},\dots ,{x}_{n}))$ and
- (4)
- $(f\xb7{x}_{0})({x}_{1},..,{x}_{n})={\sum}_{k=0}^{n-1}{(-1)}^{k}{t}_{n+1}({x}_{0},{x}_{1},\dots ,{x}_{n};{u}_{1}(n+1),{u}_{k+2}(n+1))\xb7f({x}_{0},\dots ,{x}_{k}{x}_{k+1},\dots ,{x}_{n})$$+{(-1)}^{n}(f({x}_{0},\dots ,{x}_{n-1}))\xb7{x}_{n}$

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Note**

**2.**

- (1)
- ${R}_{h}S=S{L}_{hS}$ and ${L}_{h}S=S{R}_{hS}$ for each x and h in A. Then, taking into account $(1)$ analogously to formula $(4)$ in Theorem 4, we put ${x}_{0}\xb7({L}_{{x}_{1}},{R}_{{x}_{2}})={\mathsf{t}}_{3}^{-1}({x}_{0},{x}_{1},{x}_{2})\xb7({x}_{0}{L}_{{x}_{1}}){L}_{{x}_{2}S}-{x}_{0}({L}_{{x}_{1}}{R}_{{x}_{2}})$$+{\mathsf{t}}_{3}^{-1}({x}_{0},{x}_{1},{x}_{2})\xb7({x}_{0}{R}_{{x}_{1}S}){R}_{{x}_{2}}.$Then, taking into account the multipliers ${\mathsf{t}}_{\mathsf{3}}$, this gives
- (2)
- ${x}_{0}\xb7({L}_{{x}_{1}},{R}_{{x}_{2}})={x}_{0}({L}_{{x}_{1}}{L}_{{x}_{2}S}-{L}_{{x}_{1}}{R}_{{x}_{2}}+{R}_{{x}_{1}S}{R}_{{x}_{2}})$ for all ${x}_{0},{x}_{1},{x}_{2}$ in G. Next, symmetrically, $S({x}_{0}\xb7({L}_{{x}_{1}},{R}_{{x}_{2}}))$ provides the formula for $({L}_{{y}_{1}},{R}_{{y}_{2}})\xb7{y}_{0}$ for each ${y}_{0},{y}_{1}$ and ${y}_{2}$ in G. We consider the enveloping algebra ${A}^{e}=A{\otimes}_{\mathcal{T}}{A}^{op}$. Extending these rules by $\mathcal{T}$-linearity on A and $A{\otimes}_{\mathcal{T}}{A}^{op}$ from G one supplies the tensor product $M={A}^{e}$ over $\mathcal{T}$ with the two-sided A-module structure.

**Corollary**

**1.**

**Proof.**

## 3. Products of Metagroups

**Theorem**

**8.**

- (1)
- $\mathcal{C}(G)={\prod}_{j\in J}\mathcal{C}({G}_{j})$.

**Proof.**

- (2)
- $Com(G):=\{a\in G:\forall b\in G,\phantom{\rule{3.33333pt}{0ex}}ab=ba\}=$$\{a\in G:\phantom{\rule{3.33333pt}{0ex}}a=\{{a}_{j}:\forall j\in J,{a}_{j}\in {G}_{j}\};\forall b\in G,\phantom{\rule{3.33333pt}{0ex}}b=\{{b}_{j}:\forall j\in J,{b}_{j}\in {G}_{j}\};\forall j\in J,\phantom{\rule{3.33333pt}{0ex}}{a}_{j}{b}_{j}={b}_{j}{a}_{j}\}={\prod}_{j\in J}Com({G}_{j})$,
- (3)
- ${N}_{l}(G):=\{a\in G:\phantom{\rule{3.33333pt}{0ex}}\forall b\in G,\phantom{\rule{3.33333pt}{0ex}}\forall c\in G,\phantom{\rule{3.33333pt}{0ex}}(ab)c=a(bc)\}=\{a\in G:\phantom{\rule{3.33333pt}{0ex}}a=\{{a}_{j}:\forall j\in J,{a}_{j}\in {G}_{j}\};\phantom{\rule{3.33333pt}{0ex}}\forall b\in G,\phantom{\rule{3.33333pt}{0ex}}b=\{{b}_{j}:\forall j\in J,{b}_{j}\in {G}_{j}\};\phantom{\rule{3.33333pt}{0ex}}\forall c\in G,\phantom{\rule{3.33333pt}{0ex}}c=\{{c}_{j}:\forall j\in J,{c}_{j}\in {G}_{j}\};\phantom{\rule{3.33333pt}{0ex}}\forall j\in J,\phantom{\rule{3.33333pt}{0ex}}({a}_{j}{b}_{j}){c}_{j}={a}_{j}({b}_{j}{c}_{j})\}={\prod}_{j\in J}{N}_{l}({G}_{j})$, and similarly,
- (4)
- ${N}_{m}(G)={\prod}_{j\in J}{N}_{m}({G}_{j})$ and
- (5)
- ${N}_{r}(G)={\prod}_{j\in J}{N}_{r}({G}_{j})$.This and $(8)$ of Definition 1 in Section 2 imply that
- (6)
- $N(G)={\prod}_{j\in J}N({G}_{j})$. Thus,
- (7)
- $\mathcal{C}(G):=Com(G)\cap N(G)={\prod}_{j\in J}\mathcal{C}({G}_{j})$.Let a, b, and c be in G. Then,$(ab)c=\{({a}_{j}{b}_{j}){c}_{j}:\phantom{\rule{3.33333pt}{0ex}}\forall j\in J,\phantom{\rule{3.33333pt}{0ex}}{a}_{j}\in {G}_{j},{b}_{j}\in {G}_{j},{c}_{j}\in {G}_{j}\}$$=\{{\mathsf{t}}_{3,{G}_{j}}({a}_{j},{b}_{j},{c}_{j}){a}_{j}({b}_{j}{c}_{j}):\phantom{\rule{3.33333pt}{0ex}}\forall j\in J,\phantom{\rule{3.33333pt}{0ex}}{a}_{j}\in {G}_{j},{b}_{j}\in {G}_{j},{c}_{j}\in {G}_{j}\}={\mathsf{t}}_{3,G}(a,b,c)a(bc)$, where
- (8)
- ${\mathsf{t}}_{3,G}(a,b,c)=\{{\mathsf{t}}_{3,{G}_{j}}({a}_{j},{b}_{j},{c}_{j}):\phantom{\rule{3.33333pt}{0ex}}\forall j\in J,\phantom{\rule{3.33333pt}{0ex}}{a}_{j}\in {G}_{j},{b}_{j}\in {G}_{j},{c}_{j}\in {G}_{j}\}$.

**Remark**

**2.**

- (1)
- Let A and B be two metagroups, and let $\mathcal{C}$ be a commutative group such that ${\mathcal{C}}_{m}(A)\hookrightarrow \mathcal{C}$, ${\mathcal{C}}_{m}(B)\hookrightarrow \mathcal{C}$, $\mathcal{C}\hookrightarrow \mathcal{C}(A)$ and $\mathcal{C}\hookrightarrow \mathcal{C}(B)$, where ${\mathcal{C}}_{m}(A)$ denotes a minimal subgroup in $\mathcal{C}(A)$ generated by $\{{t}_{A}(a,b,c):\phantom{\rule{3.33333pt}{0ex}}a\in A,b\in A,c\in A\}$.Using direct products, it is always possible to extend either A or B to get such a case. In particular, either A or B may be a group. Let an equivalence relation Ξ on the Cartesian product $A\times B$ be such that
- (2)
- $(\gamma v,b)\Xi (v,\gamma b)$ and $(\gamma v,b)\Xi \gamma (v,b)$ and $(\gamma v,b)\Xi (v,b)\gamma $for every v in A, b in B and γ in $\mathcal{C}$.
- (3)
- Let $\varphi :A\to \mathcal{A}(B)$ be a single-valued mapping, where $\mathcal{A}(B)$ denotes a family of all bijective surjective single-valued mappings of B onto B subjected to the conditions given below. If $a\in A$ and $b\in B$, then ${b}^{a}$ is written instead of $\varphi (a)b$ for short, where $\varphi (a):B\to B$. Also, let ${\eta}_{\varphi}:A\times A\times B\to \mathcal{C}$, ${\kappa}_{\varphi}:A\times B\times B\to \mathcal{C}$ and ${\xi}_{\varphi}:((A\times B)/\Xi )\times ((A\times B)/\Xi )\to \mathcal{C}$ be single-valued mappings written as η, κ, and ξ for short, such that
- (4)
- ${({b}^{u})}^{v}={b}^{vu}\eta (v,u,b)$, $\phantom{\rule{3.33333pt}{0ex}}{e}^{u}=e$, ${b}^{e}=b$;
- (5)
- $\eta (v,u,\gamma b)=\eta (v,u,b)$;
- (6)
- ${(cb)}^{u}={c}^{u}{b}^{u}\kappa (u,c,b)$;
- (7)
- $\kappa (u,\gamma c,b)=\kappa (u,c,\gamma b)=\kappa (u,c,b)$ and$\kappa (u,\gamma ,b)=\kappa (u,b,\gamma )=e$;
- (8)
- $\xi ((\gamma u,c),(v,b))=\xi ((u,c),(\gamma v,b))=\xi ((u,c),(v,b))$ and$\xi ((\gamma ,e),(v,b))=e$ and $\xi ((u,c),(\gamma ,e))=e$for every u and v in A, b, c in B, γ in $\mathcal{C}$, where e denotes the neutral element in $\mathcal{C}$ and in A and B.We put
- (9)
- $({a}_{1},{b}_{1})({a}_{2},{b}_{2})=({a}_{1}{a}_{2},\xi (({a}_{1},{b}_{1}),({a}_{2},{b}_{2})){b}_{1}{b}_{2}^{{a}_{1}})$for each ${a}_{1}$, ${a}_{2}$ in A, ${b}_{1}$ and ${b}_{2}$ in B.

**Theorem**

**9.**

**Proof.**

- (1)
- ${I}_{1}={\mathsf{t}}_{3}(({a}_{1},{b}_{1}),({a}_{2},{b}_{2}),({a}_{3},{b}_{3})){I}_{2}$ with
- (2)
- ${\mathsf{t}}_{3}(({a}_{1},{b}_{1}),({a}_{2},{b}_{2}),({a}_{3},{b}_{3}))={\mathsf{t}}_{3,A}({a}_{1},{a}_{2},{a}_{3}){\mathsf{t}}_{3,B}({b}_{1},{b}_{2}^{{a}_{1}},{b}_{3}^{{a}_{1}{a}_{2}})$$\xi (({a}_{1},{b}_{1}),({a}_{2}{a}_{3},{b}_{2}{b}_{3}^{{a}_{2}})){[\xi (({a}_{2},{b}_{2}),({a}_{3},{b}_{3}))]}^{{a}_{1}}\kappa ({a}_{1},{b}_{2},{b}_{3}^{{a}_{2}})\eta ({a}_{1},{a}_{2},{b}_{3})$$/[\xi (({a}_{1},{b}_{1}),({a}_{2},{b}_{2}))\xi (({a}_{1}{a}_{2},{b}_{1}{b}_{2}^{{a}_{1}}),({a}_{3},{b}_{3}))]$.

- (3)
- $({a}_{1},{b}_{1})(a,b)=(e,e)$, where $a\in A$, $b\in B$.
- (4)
- ${a}_{1}=e/a$.Consequently, $\xi ((e/a,{b}_{1}),(a,b)){b}_{1}{b}^{(e/a)}=e$, and hence
- (5)
- ${b}_{1}=e/[\xi ((e/a,{b}^{(e/a)}),(a,b)){b}^{(e/a)}]$.

- (6)
- $(a,b)({a}_{2},{b}_{2})=(e,e)$, where $a\in A$, $b\in B$ we infer that
- (7)
- ${a}_{2}=a\backslash e$.Consequently, $\xi ((a,b),(a\backslash e,{b}_{2}))b{b}_{2}^{a}=e$, and hence, ${b}_{2}^{a}=[\xi ((a,b),(a\backslash e,{b}_{2}))b]\backslash e$. On the other hand, ${({b}_{2}^{a})}^{e/a}=\eta (e/a,a,{b}_{2}){b}_{2}$ Consequently,
- (8)
- ${b}_{2}={(b\backslash e)}^{e/a}/\{[{(\xi ((a,b),(a\backslash e,{(b\backslash e)}^{e/a}))]}^{e/a}\eta (e/a,a,{(b\backslash e)}^{e/a})\}$.

- (9)
- $(a,b)\backslash (c,d)=((a,b)\backslash (e,e))(c,d)$${\mathsf{t}}_{3}((e,e)/(a,b),(a,b),((a,b)\backslash (e,e))(c,d))/{\mathsf{t}}_{3}((e,e)/(a,b),(a,b),(a,b)\backslash (e,e))$;
- (10)
- $(c,d)/(a,b)=(c,d)((e,e)/(a,b))$${\mathsf{t}}_{3}((e,e)/(a,b),(a,b),(a,b)\backslash (e,e))/{\mathsf{t}}_{3}((c,d)(e/(a,b)),(a,b),(a,b)\backslash (e,e))$

**Definition**

**5.**

**Remark**

**3.**

**Conclusions**

**1.**

## Funding

## Conflicts of Interest

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Ludkowski, S.V.
Cohomology Theory of Nonassociative Algebras with Metagroup Relations. *Axioms* **2019**, *8*, 78.
https://doi.org/10.3390/axioms8030078

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Ludkowski SV.
Cohomology Theory of Nonassociative Algebras with Metagroup Relations. *Axioms*. 2019; 8(3):78.
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Ludkowski, Sergey V.
2019. "Cohomology Theory of Nonassociative Algebras with Metagroup Relations" *Axioms* 8, no. 3: 78.
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