# Actions and semi-direct products in categories of groups with action

Tamar Datuashvili Corresponding author<EMAIL_ADDRESS>(T. Datuashvili)
A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State
University, 6 Tamarashvii Str., Tbilisi 0177, Georgia Tunçar Þahan  Department
of Mathematics, Aksaray University, Aksaray, Turkey

###### Abstract

Derived actions in the category of groups with action on itself
$\mathbf{Gr}^{\bullet}$ are defined and described. This category plays a
crucial role in the solution of Loday’s two problems stated in the literature.
A full subcategory of reduced groups with action $\mathbf{rGr}^{\bullet}$ of
$\mathbf{Gr}^{\bullet}$ is introduced, which is not a category of interest but
has some properties, which can be applied in the investigation of action
representability in this category; these properties are similar to those,
which were used in the construction of universal strict general actors in the
category of interest. Semi-direct product constructions are given in
$\mathbf{Gr}^{\bullet}$ and $\mathbf{rGr}^{\bullet}$ and it is proved that an
action is a derived action in $\mathbf{Gr}^{\bullet}$ (resp.
$\mathbf{rGr}^{\bullet}$) if and only if the corresponding semi-direct product
is and object of $\mathbf{Gr}^{\bullet}$ (resp. $\mathbf{rGr}^{\bullet}$). The
results obtained in this paper will be applied in the forthcoming paper on the
representability of actions in the category $\mathbf{rGr}^{\bullet}$.
Mathematics Subject Classification (2020): 08A99, 08C05, 22F05.
Keywords: Group with action, semi-direct product, action, extension

##  1 Introduction

We develop action theory in the category of groups with action on itself
$\mathbf{Gr}^{\bullet}$, introduced in [5, 6, 7], where it played a main role
in the solution of Loday’s two problems stated in [9, 10]. This category is
neither a category of interest in the sense of Orzech [11, 12], nor a modified
category of interest [2]. It is a category of groups with operations, but
doesn’t satisfy all conditions stated in [13]. The category
$\mathbf{Gr}^{\bullet}$ is a category of $\Omega$-groups in the sense of
Kurosh [8]. Actions are defined in $\mathbf{Gr}^{\bullet}$ as derived actions
from split extensions in this category as it is in the category of interest or
in any semi-abelian category [1]. We describe derived action conditions in
this category and construct a semi-direct product $B\ltimes A$, where
$A,B\in\mathbf{Gr}^{\bullet}$ and $B$ has a derived action on $A$. We prove
that an action of $B$ on $A$ is a derived action if and only if $B\ltimes
A\in\mathbf{Gr}^{\bullet}$ (Theorem 3.2). Then we define a full subcategory
$\mathbf{rGr}^{\bullet}$ in $\mathbf{Gr}^{\bullet}$ and describe derived
actions in $\mathbf{rGr}^{\bullet}$. Our interest is to investigate the
existence of a universal acting object on an object
$G\in\mathbf{Gr}^{\bullet}$ applying the results obtained in [3, 4] for
categories of interest. Since the category $\mathbf{Gr}^{\bullet}$ is far from
being category of interest we found its subcategory $\mathbf{rGr}^{\bullet}$,
which is not a category of interest, but has interesting properties which are
close to those ones we use in the construction of a universal strict general
actor for any object of a category of interest in [3, 4]. We prove necessary
and sufficient condition for the action of $B$ on $A$,
$A,B\in\mathbf{rGr}^{\bullet}$, to be a derived action in terms of the semi-
direct product $B\ltimes A$, like we have in $\mathbf{Gr}^{\bullet}$ (Theorem
4.2). Applying the results of this paper, we will prove that under certain
conditions on the object $G\in\mathbf{rGr}^{\bullet}$, it has representable
actions in the sense of [1], i.e. a universal acting object, which represents
all actions on $A$.

##  2 Preliminary definitions and results

Let $G$ be a group which acts on itself from the right side, i.e. we have a
map $\varepsilon\colon G\times G\rightarrow G$ with

$\displaystyle\varepsilon(g,g^{\prime}+g^{\prime\prime})$
$\displaystyle=\varepsilon(\varepsilon(g,g^{\prime}),g^{\prime\prime})$
$\displaystyle\varepsilon(g,0)$ $\displaystyle=g$
$\displaystyle\varepsilon(g^{\prime}+g^{\prime\prime},g)$
$\displaystyle=\varepsilon(g^{\prime}+g)+\varepsilon(g^{\prime\prime}+g)$

for $g,g^{\prime},g^{\prime\prime}\in G$. Denote $\varepsilon(g,h)=g^{h}$, for
$g,h\in G$.

We denote the group operation additively, nevertheless the group is not
commutative in general. From the third condition on $\varepsilon$ it follows
that

$0^{h}=0,\text{ for any }h\in G.$

If $(G^{\prime},\varepsilon^{\prime})$ is another group with action then a
homomorphism $(G,\varepsilon)\rightarrow(G^{\prime},\varepsilon^{\prime})$ is
a group homomorphism $\varphi\colon G\rightarrow G^{\prime}$, for which the
diagram

$\textstyle{G\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varphi\times\varphi}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{G^{\prime}\times
G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\prime}}$$\textstyle{G^{\prime}}$

commutes. In other words, we have

$\varphi\left(g^{h}\right)={\varphi(g)}^{\varphi(h)}$ (2.1)

for all $g,h\in G$.

Note that action defined above is a split derived action in the sense of [11,
12].

According to Kurosh [8] an $\Omega$-group is a group with a system of $n$-ary
algebraic operations $\Omega_{n\geq 0}$, which satisfy the condition

$000\cdots 0\omega=0,$ (2.2)

where 0 is the identity element of $G$, and 0 on the left side occurs $n$
times if $\omega$ is an $n$-ary operation. In special cases $\Omega$-groups
give groups, rings, associative and non-associative algebras like Lie and
Leibniz algebras etc. and groups with action on itself as well. In the latter
case $\Omega$ consists of one binary operation which is an action or $\Omega$
consists of only unary operations, which are elements of $G$, and this
operation is an action again. In both cases condition (2.2) is satisfied.
Denote the category of groups with action on itself by
$\mathbf{Gr}^{\bullet}$; here the action is considered as a binary operation
and morphisms between the objects in $\mathbf{Gr}^{\bullet}$ are group
homomorphisms satisfying condition (2.1).

Let $G\in\mathbf{Gr}^{\bullet}$.

######  Definition 2.1.

[5] A non-empty subset $A$ of $G$ is called an ideal of $G$ if it satisfies
the following conditions

  1. (1)

$A$ is a normal subgroup of $G$ as a group;

  2. (2)

$a^{g}\in A$, for any $a\in A$ and $g\in G$;

  3. (3)

$-g+g^{a}\in A$, for any $a\in A$ and $g\in G$.

Note that the condition (3) in this definition is equivalent to the condition,
that $g^{a}-g\in A$, since $(-g)^{a}=-g^{a},$ for any $a\in A$ and $g\in G$.
This definition is equivalent to the definition of an ideal given in [8] for
$\Omega$-groups in the case where $\Omega$ consists of one binary operation of
action, one can see the proof in [5].

##  3 Actions and semi-direct products in $\mathbf{Gr}^{\bullet}$

Let $A,B\in\mathbf{Gr}^{\bullet}$. An action of $B$ on $A$ by definition is a
triple of mappings
$\beta=(\beta_{+},\beta_{\ast},\beta_{{\ast}^{\circ}})\colon B\times
A\rightarrow A$, where $\ast$ is a binary operation of action,
${\ast}^{\circ}$ is its dual operation in $\mathbf{Gr}^{\bullet}$, i.e.
$\beta_{+}(b,a)=b\cdot a$, $\beta_{\ast}(b,a)=a\ast b={a}^{b}$ and
$\beta_{{\ast}^{\circ}}(b,a)=a{\ast}^{\circ}b={b}^{a}$.

In the category of interest or category of groups with operations there is a
condition $0\ast g=g\ast 0=0$, for any binary operation
$\ast\in\Omega\backslash\\{+\\}$, any object $G$ in this category and any
element $g\in G$. In the category $\mathbf{Gr}^{\bullet}$ we have ${0}^{g}=0$,
for any $G\in\mathbf{Gr}^{\bullet}$ and any $g\in G$, but ${g}^{0}\neq 0$ in
general. Therefore we modify the definition of derived action due to split
extensions [11, 12, 13], known for the category of groups with operations or
category of interest, for the category $\mathbf{Gr}^{\bullet}$. Note that the
definition of derived action from the split extension agrees with the
definition of action in a semi-abelian category [1].

Let $A,B\in\mathbf{Gr}^{\bullet}$. An extension of $B$ by $A$ is a sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(3.1)

in which $p$ is surjective and $i$ is the kernel of $p$. We say that an
extension is split if there is a morphism $j\colon B\rightarrow E$, such that
$pj=1_{B}$. We will identify $i(a)$ with $a$.

A split extension induces a triple of actions of $B$ on $A$ corresponding to
the operation of addition, action and its dual operation in
$\mathbf{Gr}^{\bullet}$. From the split extension (3.1) for any $b\in B$ and
$a\in A$ we define

$\displaystyle b\cdot a$ $\displaystyle=j(b)+a-j(b)$ (3.2)
$\displaystyle{b}^{a}$ $\displaystyle={j(b)}^{a}-j(b)$ (3.3)
$\displaystyle{a}^{b}$ $\displaystyle={a}^{j(b)}$ (3.4)

Actions defined by (3.2)-(3.4) will be called derived actions of $B$ on $A$ as
it is in the case of groups with operations or category of interest. Note that
(3.3) differs from what we have in the noted known cases, since as we have
mentioned above ${b}^{0}\neq 0$ in $B$.

######  Proposition 3.1.

Let $A,B,\in\mathbf{Gr}^{\bullet}$. Derived actions of $B$ on $A$ satisfy the
following conditions:

  1. (a)

well-known group action conditions for the dot left action:

$\displaystyle b\cdot(a_{1}+a_{2})$ $\displaystyle=b\cdot a_{1}+b\cdot a_{2}$
$\displaystyle(b_{1}+b_{2})\cdot a$ $\displaystyle=b_{1}\cdot(b_{2}\cdot a)$
$\displaystyle 0\cdot a$ $\displaystyle=a$

where $a,a_{1},a_{2}\in A$ and $b,b_{1},b_{2}\in B$,

  2. (b)

${0_{A}}^{b}=0_{A}$, ${0_{B}}^{a}=0_{A}$, ${b}^{0_{A}}=0_{A}$, ${a}^{0_{B}}=a$
where $0_{A}$ and $0_{B}$ denote the zero elements of $A$ and $B$
respectively. For any $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$,

    1. $\bm{(1_{A})}$

${(a+a^{\prime})}^{b}={a}^{b}+{(a^{\prime})}^{b}$,

    2. $\bm{(2_{A})}$

${(b+b^{\prime})}^{a}={b}^{a}+b\cdot\left((b^{\prime})^{a}\right)$,

    3. $\bm{(3_{A})}$

${(b\cdot
a)}^{a^{\prime}}+{b}^{a^{\prime}}={b}^{a^{\prime}}+b\cdot\left({a}^{a^{\prime}}\right)$,

    4. $\bm{(4_{A})}$

$\left(b\cdot a\right)^{b^{\prime}}={b}^{b^{\prime}}\cdot{a}^{b^{\prime}}$,

    5. $\bm{(1_{B})}$

${b}^{(a+a^{\prime})}=\left({b}^{a}\right)^{a^{\prime}}+{b}^{a^{\prime}}$,

    6. $\bm{(2_{B})}$

${a}^{b+b^{\prime}}=\left({a}^{b}\right)^{b^{\prime}}$

    7. $\bm{(3_{B})}$

$\left({a}^{(b\cdot
a^{\prime})}\right)^{b}=\left({a}^{b}\right)^{a^{\prime}}$,

    8. $\bm{(4_{B})}$

$\left({b}^{(b^{\prime}\cdot
a)}\right)^{b^{\prime}}=\left({b}^{b^{\prime}}\right)^{a}$.

Note that as it will be shown in the proof of Theorem 3.2, all properties
noted in (b) except ${a}^{0_{B}}=a$ follow from $\bm{(1_{A})}$, $\bm{(2_{A})}$
and $\bm{(1_{B})}$. Nevertheless we preferred for explicitness to state in the
theorem these properties separately.

###### Proof.

  1. (a)

This is obvious.

  2. (b)

Follows from the action properties $({0_{E}}^{a}=0_{E},{a}^{0_{E}}=a,\text{
for all }a\in A)$ and the definition of the derived action corresponding to
the action operation and its dual (3.3) and (3.4).

    1. $\bm{(1_{A})}$

Let $a,a^{\prime}\in A$ and $b\in B$; then

$\displaystyle{(a+a^{\prime})}^{b}$ $\displaystyle={(a+a^{\prime})}^{j(b)}$
$\displaystyle={a}^{j(b)}+{(a^{\prime})}^{j(b)}$
$\displaystyle={a}^{b}+{(a^{\prime})}^{b}$

    2. $\bm{(2_{A})}$

Let $a\in A$ and $b,b^{\prime}\in B$; then

$\displaystyle{(b+b^{\prime})}^{a}$
$\displaystyle=\left(j(b+b^{\prime})\right)^{a}-j\left(b+b^{\prime}\right)$
$\displaystyle=j\left(b\right)^{a}+j\left(b^{\prime}\right)^{a}-j\left(b^{\prime}\right)-j\left(b\right)$
$\displaystyle=j\left(b\right)^{a}-j\left(b\right)+j\left(b\right)+j\left(b^{\prime}\right)^{a}-j\left(b^{\prime}\right)-j\left(b\right)$
$\displaystyle={b}^{a}+b\cdot\left((b^{\prime})^{a}\right)$

    3. $\bm{(3_{A})}$

Let $a,a^{\prime}\in A$ and $b\in B$; then

$\displaystyle{(b\cdot a)}^{a^{\prime}}+{b}^{a^{\prime}}$
$\displaystyle=\left(j(b)+a-j(b)\right)^{a^{\prime}}+{j(b)}^{a^{\prime}}-j(b)$
$\displaystyle=j(b)^{a^{\prime}}+a^{a^{\prime}}-j(b)^{a^{\prime}}+{j(b)}^{a^{\prime}}-j(b)$
$\displaystyle=j(b)^{a^{\prime}}-j(b)+{j(b)}+a^{a^{\prime}}-j(b)$
$\displaystyle={b}^{a^{\prime}}+b\cdot\left({a}^{a^{\prime}}\right)$

    4. $\bm{(4_{A})}$

Let $a\in A$ and $b,b^{\prime}\in B$; then

$\displaystyle\left(b\cdot a\right)^{b^{\prime}}$
$\displaystyle=\left(j(b)+a-j(b)\right)^{b^{\prime}}$
$\displaystyle=j(b)^{j(b^{\prime})}+a^{j(b^{\prime})}-j(b)^{j(b^{\prime})}$
$\displaystyle=j\left(b^{b^{\prime}}\right)+a^{j(b^{\prime})}-j\left(b^{b^{\prime}}\right)$
$\displaystyle={b}^{b^{\prime}}\cdot{a}^{b^{\prime}}$

    5. $\bm{(1_{B})}$

Let $a,a^{\prime}\in A$ and $b\in B$; then

$\displaystyle{b}^{(a+a^{\prime})}$
$\displaystyle={j(b)}^{(a+a^{\prime})}-j(b)$
$\displaystyle={\left({j(b)}^{a}\right)}^{a^{\prime}}-{j(b)}^{a^{\prime}}+{j(b)}^{a^{\prime}}-j(b)$
$\displaystyle={\left({j(b)}^{a}-{j(b)}\right)}^{a^{\prime}}+{j(b)}^{a^{\prime}}-j(b)$
$\displaystyle=\left({b}^{a}\right)^{a^{\prime}}+{b}^{a^{\prime}}$

    6. $\bm{(2_{B})}$

Let $a\in A$ and $b,b^{\prime}\in B$; then

$\displaystyle{a}^{b+b^{\prime}}$ $\displaystyle={a}^{j(b)+j(b^{\prime})}$
$\displaystyle=\left({a}^{b}\right)^{b^{\prime}}$

    7. $\bm{(3_{B})}$

Let $a,a^{\prime}\in A$ and $b\in B$; then

$\displaystyle\left({a}^{(b\cdot a^{\prime})}\right)^{b}$
$\displaystyle=\left({a}^{(j(b)+a^{\prime}-j(b))}\right)^{j(b)}$
$\displaystyle={a}^{j(b)+a^{\prime}}$
$\displaystyle=\left({a}^{j(b)}\right)^{a^{\prime}}$
$\displaystyle=\left({a}^{b}\right)^{a^{\prime}}$

    8. $\bm{(4_{B})}$

Let $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$; then

$\displaystyle\left({b}^{(b^{\prime}\cdot a)}\right)^{b^{\prime}}$
$\displaystyle=\left({j(b)}^{(j(b^{\prime})+a-j(b^{\prime}))}-j(b)\right)^{j(b^{\prime})}$
$\displaystyle=\left(\left(\left({j(b)}^{j(b^{\prime})}\right)^{a}\right)^{-j(b^{\prime})}\right)^{j(b^{\prime})}-{j(b)}^{j(b^{\prime})}$
$\displaystyle=\left({j(b)}^{j(b^{\prime})}\right)^{a}-{j(b)}^{j(b^{\prime})}$
$\displaystyle=\left(j\left({b}^{b^{\prime}}\right)\right)^{a}-j\left({b}^{b^{\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime}}\right)^{a}$

∎

Given a triple of actions of $B$ on $A$ in $\mathbf{Gr}^{\bullet}$, we can
define operations on the product $B\times A$ in the following way:

$\displaystyle(b,a)+(b^{\prime},a^{\prime})$
$\displaystyle=(b+b^{\prime},a+b\cdot a^{\prime})$ (3.5)
$\displaystyle{(b,a)}^{(b^{\prime},a^{\prime})}$
$\displaystyle=({b}^{b^{\prime}},({a}^{a^{\prime}})^{b^{\prime}}+({b}^{a^{\prime}})^{b^{\prime}})$
(3.6)

for any $(b,a),(b^{\prime},a^{\prime})\in B\times A$. This kind of universal
algebra will be callaed semi-direct product and denoted by $B\ltimes A$.

######  Theorem 3.2.

Let $A,B\in\mathbf{Gr}^{\bullet}$, if
$\beta=(\beta_{+},\beta_{\ast},\beta_{{\ast}^{\circ}})$ is a triple of actions
of $B$ on $A$, then the following conditions are equivalent:

  1. (1)

$\beta$ is a triple of derived actions of $B$ on $A$.

  2. (2)

$\beta_{+}$ satisfies group action conditions, $\beta$ satisfies conditions
$\bm{(1_{A})}-\bm{(4_{A})}$, $\bm{(1_{B})}-\bm{(4_{B})}$ and the condition
$a^{0_{B}}=a$, for any $a\in A$.

  3. (3)

The semi-direct product $B\ltimes A$ is an object in $\mathbf{Gr}^{\bullet}$.

###### Proof.

  1. (1)$\Rightarrow$(2):

by Proposition 3.1.

  2. (2)$\Rightarrow$(3):

First of all we will show that from $\bm{(1_{A})}$, $\bm{(2_{A})}$ and
$\bm{(1_{B})}$ follow the conditions of (b) except the one ${a}^{0_{B}}=a$.
From $\bm{(1_{A})}$ we have

${0_{A}}^{b}={0_{A}+0_{A}}^{b}={0_{A}}^{b}+{0_{A}}^{b}$

and then ${0_{A}}^{b}=0_{A}$.

From $\bm{(2_{A})}$ we have

${0_{B}}^{a}={0_{B}+0_{B}}^{a}={0_{B}}^{a}+{0_{B}}\cdot\left({0_{B}}^{b}\right)={0_{B}}^{a}+{0_{B}}^{a}$

and then ${0_{B}}^{a}=0_{A}$. Note that we will not use this property in the
proof of (2)$\Rightarrow$(3).

From $\bm{(1_{B})}$ we have

${b}^{0_{A}}={b}^{0_{A}+0_{A}}=\left({b}^{0_{A}}\right)^{0_{A}}+{b}^{0_{A}}$

since $a^{0_{A}}=a$ for any $a\in A$, we obtain

${b}^{0_{A}}={b}^{0_{A}}+{b}^{0_{A}}$

and then ${b}^{0_{A}}=0_{A}$.

Now we shall prove that the semi-direct product $B\ltimes
A\in\mathbf{Gr}^{\bullet}$. Obviously, $B\ltimes A$ is a group as it is in the
case of groups. We have to show the following equalities for any
$(b,a),(b^{\prime},a^{\prime}),(b^{\prime\prime},a^{\prime\prime})\in B\ltimes
A$:

    1. (a)

${(b,a)}^{(b^{\prime},a^{\prime})+(b^{\prime\prime},a^{\prime\prime})}=\left({(b,a)}^{(b^{\prime},a^{\prime})}\right)^{(b^{\prime\prime},a^{\prime\prime})}$

    2. (b)

$\left({(b,a)}+{(b^{\prime},a^{\prime})}\right)^{(b^{\prime\prime},a^{\prime\prime})}={(b,a)}^{(b^{\prime\prime},a^{\prime\prime})}+{(b^{\prime},a^{\prime})}^{(b^{\prime\prime},a^{\prime\prime})}$

    3. (c)

${(b,a)}^{(0_{B},0_{A})}={(b,a)}$.

First we prove the equality in (a).

    1. (a)

We have

$\displaystyle{(b,a)}^{(b^{\prime},a^{\prime})+(b^{\prime\prime},a^{\prime\prime})}$
$\displaystyle={(b,a)}^{(b^{\prime}+b^{\prime\prime},a^{\prime}+b^{\prime}\cdot
a^{\prime\prime})}$
$\displaystyle=\left({b}^{b^{\prime}+b^{\prime\prime}},\left({a}^{a^{\prime}+b^{\prime}\cdot
a^{\prime\prime}}\right)^{b^{\prime}+b^{\prime\prime}}+\left({b}^{a^{\prime}+b^{\prime}\cdot
a^{\prime\prime}}\right)^{b^{\prime}+b^{\prime\prime}}\right)$
$\displaystyle=\left(\left({b}^{b^{\prime}}\right)^{b^{\prime\prime}},\left(\left(\left({a}^{a^{\prime}}\right)^{b^{\prime}\cdot
a^{\prime\prime}}\right)^{b^{\prime}}\right)^{b^{\prime\prime}}+\left(\left({b}^{a^{\prime}}\right)^{b^{\prime}\cdot
a^{\prime\prime}}+{b}^{b^{\prime}\cdot
a^{\prime\prime}}\right)^{b^{\prime}+b^{\prime\prime}}\right)$
$\displaystyle=\left(\left({b}^{b^{\prime}}\right)^{b^{\prime\prime}},\left(\left(\left({a}^{a^{\prime}}\right)^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left(\left({b}^{a^{\prime}}\right)^{b^{\prime}\cdot
a^{\prime\prime}}\right)^{b^{\prime}}\right)^{b^{\prime\prime}}+\left(\left({b}^{b^{\prime}\cdot
a^{\prime\prime}}\right)^{b^{\prime}}\right)^{b^{\prime\prime}}\right)$
$\displaystyle=\left(\left({b}^{b^{\prime}}\right)^{b^{\prime\prime}},\left(\left(\left({a}^{a^{\prime}}\right)^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left(\left({b}^{a^{\prime}}\right)^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left({b}^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right).$

On the other hand

$\displaystyle\left({(b,a)}^{(b^{\prime},a^{\prime})}\right)^{(b^{\prime\prime},a^{\prime\prime})}$
$\displaystyle=\left({b}^{b^{\prime}},\left({a}^{a^{\prime}}\right)^{b^{\prime}}+\left({b}^{a^{\prime}}\right)^{b^{\prime}}\right)^{(b^{\prime\prime},a^{\prime\prime})}$
$\displaystyle=\left(\left({b}^{b^{\prime}}\right)^{b^{\prime\prime}},\left(\left(\left({a}^{a^{\prime}}\right)^{b^{\prime}}+\left({b}^{a^{\prime}}\right)^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left({b}^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$
$\displaystyle=\left(\left({b}^{b^{\prime}}\right)^{b^{\prime\prime}},\left(\left(\left({a}^{a^{\prime}}\right)^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left(\left({b}^{a^{\prime}}\right)^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left({b}^{b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$

From which we conclude that condition (a) holds in $B\ltimes A$.

Now we check condition (b).

    1. (b)

We have

$\displaystyle\left({(b,a)}+{(b^{\prime},a^{\prime})}\right)^{(b^{\prime\prime},a^{\prime\prime})}$
$\displaystyle=\left(b+b^{\prime},a+b\cdot
a^{\prime}\right)^{(b^{\prime\prime},a^{\prime\prime})}$
$\displaystyle=\left(\left(b+b^{\prime}\right)^{b^{\prime\prime}},\left(\left(a+b\cdot
a^{\prime}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left(b+b^{\prime}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$
$\displaystyle=\left(b^{b^{\prime\prime}}+{b^{\prime}}^{b^{\prime\prime}},\left({a}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(\left(b\cdot
a^{\prime}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left({b}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(b\cdot\left({b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$
$\displaystyle=\left(b^{b^{\prime\prime}}+{b^{\prime}}^{b^{\prime\prime}},\left({a}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left({b}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left(b\cdot\left({a^{\prime}}^{a^{\prime\prime}}\right)\right)^{b^{\prime\prime}}+\left(b\cdot\left(b^{\prime}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$

Here we apply condition $\bm{(3_{A})}$. We have the following equalities.

$\displaystyle{(b,a)}^{(b^{\prime\prime},a^{\prime\prime})}+{(b^{\prime},a^{\prime})}^{(b^{\prime\prime},a^{\prime\prime})}$
$\displaystyle=\left({b}^{b^{\prime\prime}},\left({a}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left({b}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)+\left({b^{\prime}}^{b^{\prime\prime}},\left({a^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left({b^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime\prime}}+{b^{\prime}}^{b^{\prime\prime}},\left({a}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left({b}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+{b}^{b^{\prime\prime}}\cdot\left({a^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+{b}^{b^{\prime\prime}}\cdot\left({b^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)$

Applying condition $\bm{(4_{A})}$ we obtain

$b\cdot\left({a^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}={b}^{b^{\prime\prime}}\cdot\left({a^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}$

and

$\left(b\cdot\left({b^{\prime}}\right)^{a^{\prime\prime}}\right)^{b^{\prime\prime}}={b}^{b^{\prime\prime}}\cdot\left({b^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}$

which proves that we have condition (b).

Finally, we check condition (c).

    1. (c)

Here, if we apply the equalities

$\left({a}^{0_{A}}\right)^{0_{B}}={a}^{0_{B}}=a$

and

$\left({b}^{0_{A}}\right)^{0_{B}}={0_{A}}^{0_{B}}=0_{A},$

then we get

${(b,a)}^{(0_{B},0_{A})}=\left({b}^{0_{B}},\left({a}^{0_{A}}\right)^{0_{B}}+\left({b}^{0_{A}}\right)^{0_{B}}\right)={(b,a)}.$

  3. (3)$\Rightarrow$(1):

Suppose $B\ltimes A\in\mathbf{Gr}^{\bullet}$, then we have a split extension

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B\ltimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{0}$
(3.7)

where $p(b,a)=b$, $i(a)=(0,a)$ and $j(b)=(b,0)$. Define derived actions from
this extension in a usual way.

$\displaystyle b~{}\overline{\cdot}~{}a$ $\displaystyle=j(b)+a-j(b)$
$\displaystyle=(b,0)+(0,a)-(b,0)$ $\displaystyle=(b,b\cdot a)+(-b,0)$
$\displaystyle=(0,b\cdot a),$

therefore the derived action corresponding to the addition operation coincides
with the given action.

Action corresponding to the action operation, denoted by $\ast$, is defined by

$\displaystyle a*b$
$\displaystyle=\left(0_{B},a\right)^{\left(b,0_{A}\right)}$
$\displaystyle=\left({0_{B}}^{b},\left({a}^{0_{A}}\right)^{b}+\left({0_{B}}^{0_{A}}\right)^{b}\right)$
$\displaystyle=\left(0,{a}^{b}\right).$

As we see this action also coincides with the given action.

For the dual to $\ast$ operation, i.e. dual action we have

$\displaystyle a{\ast}^{\circ}b$
$\displaystyle=\left(b,0_{A}\right)^{\left(0_{B},a\right)}-\left(b,0_{A}\right)$
$\displaystyle=\left(b^{0_{B}},\left({0_{A}}^{a}\right)^{0_{B}}+\left({b}^{a}\right)^{0_{B}}\right)-\left(b,0_{A}\right)$
$\displaystyle=\left(b,b^{a}\right)-\left(b,0_{A}\right)$
$\displaystyle=\left(b-b,b^{a}+b\cdot 0_{A}\right)$
$\displaystyle=\left(0_{B},b^{a}\right).$

Therefore this action also coincides with the given action of $B$ on $A$,
which proves that the given action of $B$ on $A$ is a derived action, which
concludes the proof of the theorem.

∎

For the examples of derived actions in the category $\mathbf{Gr}^{\bullet}$
see Section 4, Lemma 4.3 and Corollary 4.4.

##  4 The subcategory
$\mathbf{rGr}^{\bullet}\hookrightarrow\mathbf{Gr}^{\bullet}$

Consider the objects $A\in\mathbf{Gr}^{\bullet}$ which satisfy two conditions:

  1. (1)

${x}^{y}+z=z+{x}^{y}$, $y\neq 0$ and

  2. (2)

${x}^{\left({y}^{z}\right)}={x}^{y}$,

for any $x,y,z\in A.$ This kind of objects will be called reduced groups with
action, and the corresponding full subcategory of $\mathbf{Gr}^{\bullet}$ will
be denoted by $\mathbf{rGr}^{\bullet}.$

Derived actions are defined in $\mathbf{rGr}^{\bullet}$ in analogous way as it
is in $\mathbf{Gr}^{\bullet}$.

######  Example 4.1.

For any set $X$ let $F(X)$ be a free group with action on itself in
$\mathbf{Gr}^{\bullet}$; one can see the construction in [6]. Let $R$ be a
congruence relation on $F(X)$ generated by the relations

${x}^{y}+z\sim z+{x}^{y}$

for any $y\neq 0$ and

${x}^{\left({y}^{z}\right)}\sim{x}^{y}$

for any $x,y,z\in F(X)$. Then the quotient object $\faktor{F(X)}{R}$ by the
$R$ obviously is an object of $\mathbf{rGr}^{\bullet}$.

######  Theorem 4.2.

Let $A,B\in\mathbf{rGr}^{\bullet}$ and
$\beta=(\beta_{+},\beta_{\ast},\beta_{{\ast}^{\circ}})\colon B\times
A\rightarrow A$ be a triple of actions of $B$ on $A$ in
$\mathbf{rGr}^{\bullet}$. Then the following conditions are equivalent:

  1. (1)

$\beta$ is a triple of derived actions in $\mathbf{rGr}^{\bullet}$.

  2. (2)

$\beta$ satisfies condition (2) of Theorem 3.2 and the following conditions

$\begin{array}[]{rclcrcl}b\cdot{a}^{a^{\prime}}&=&{a}^{a^{\prime}}&&{a}^{b}+a^{\prime}&=&a^{\prime}+{a}^{b}\\\
b\cdot{a}^{b^{\prime}}&=&{a}^{b^{\prime}}&&{a}^{\left({a^{\prime}}^{b}\right)}&=&{a}^{a^{\prime}}\\\
{b}^{b^{\prime}}\cdot a&=&a&&{a}^{\left({b}^{a^{\prime}}\right)}&=&a\\\
{b}^{\left({a}^{a^{\prime}}\right)}&=&{b}^{a}&&{b}^{\left({b^{\prime}}^{a}\right)}&=&0\\\
{a}^{\left({b}^{b^{\prime}}\right)}&=&{a}^{b}&&{b}^{\left({a}^{b^{\prime}}\right)}&=&{b}^{a}\end{array}$
(4.1)

for any $a,a^{\prime}\in A$, $b,b^{\prime}\in B$. Note that under the
conditions (4.1), $\bm{(2_{A})}$, $\bm{(3_{A})}$ and $\bm{(4_{A})}$ have
simpler forms.

  3. (3)

The semi-direct product $B\ltimes A$ is an object in $\mathbf{rGr}^{\bullet}$.

###### Proof.

  1. (1)$\Rightarrow$(2):

We will check only the conditions ${a}^{\left({b}^{a^{\prime}}\right)}=a$,
${b}^{\left({b^{\prime}}^{a}\right)}=0$ and
${b}^{\left({a}^{b^{\prime}}\right)}={b}^{a}$. Other conditions are obvious.

    1. (i)

${a}^{\left({b}^{a^{\prime}}\right)}={a}^{\left({j(b)}^{a^{\prime}}\right)-j(b)}={a}^{{j(b)}-j(b)}={a}^{0}=a$;

    2. (ii)

${b}^{\left({b^{\prime}}^{a}\right)}={j(b)}^{\left({j(b^{\prime})}^{a}-j(b^{\prime})\right)}-j(b)=\left({j(b)}^{j(b^{\prime})}\right)^{-j(b^{\prime})}-j(b)={j(b)}^{0}-j(b)=0$;

    3. (iii)

${b}^{\left({a}^{b^{\prime}}\right)}={j(b)}^{\left({a}^{j(b^{\prime})}\right)}-j(b)={j(b)}^{a}-j(b)={b}^{a}$.

  2. (2)$\Rightarrow$(3):

By Theorem 3.2 we need to prove only that

${(b,a)}^{(b^{\prime},a^{\prime})}+(b^{\prime\prime},a^{\prime\prime})=(b^{\prime\prime},a^{\prime\prime})+{(b,a)}^{(b^{\prime},a^{\prime})}$

and

${(b,a)}^{\left({(b^{\prime},a^{\prime})}^{(b^{\prime\prime},a^{\prime\prime})}\right)}={(b,a)}^{(b^{\prime},a^{\prime})}$

for any $(b,a),(b^{\prime},a^{\prime}),(b^{\prime\prime},a^{\prime\prime})\in
B\ltimes A$. We have

$\displaystyle{(b,a)}^{(b^{\prime},a^{\prime})}+(b^{\prime\prime},a^{\prime\prime})$
$\displaystyle=\left({b}^{b^{\prime}},{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}\right)+(b^{\prime\prime},a^{\prime\prime})$
$\displaystyle=\left({b}^{b^{\prime}}+b^{\prime\prime},{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}+{b}^{b^{\prime}}\cdot
a^{\prime\prime}\right)$
$\displaystyle=\left({b}^{b^{\prime}}+b^{\prime\prime},{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}+a^{\prime\prime}\right).$

On the other hand

$\displaystyle(b^{\prime\prime},a^{\prime\prime})+{(b,a)}^{(b^{\prime},a^{\prime})}$
$\displaystyle=(b^{\prime\prime},a^{\prime\prime})+\left({b}^{b^{\prime}},{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}\right)$
$\displaystyle=\left(b^{\prime\prime}+{b}^{b^{\prime}},a^{\prime\prime}+b^{\prime\prime}\cdot{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+b^{\prime\prime}\cdot{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime}}+b^{\prime\prime},a^{\prime\prime}+{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime}}+b^{\prime\prime},{\left({a}^{a^{\prime}}\right)}^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}+a^{\prime\prime}\right).$

which proves the first identity. For the second identity we have

$\displaystyle{(b,a)}^{\left({(b^{\prime},a^{\prime})}^{(b^{\prime\prime},a^{\prime\prime})}\right)}$
$\displaystyle={(b,a)}^{\left({b^{\prime}}^{b^{\prime\prime}},\left({a^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}+\left({b^{\prime}}^{a^{\prime\prime}}\right)^{b^{\prime\prime}}\right)}$
$\displaystyle=\left({b}^{\left({b^{\prime}}^{b^{\prime\prime}}\right)},{\left({a}^{\left({\left({a^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}+{\left({b^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}\right)}^{\left({b^{\prime}}^{b^{\prime\prime}}\right)}+{\left({b}^{\left({\left({a^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}+{\left({b^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}\right)}^{\left({b^{\prime}}^{b^{\prime\prime}}\right)}\right)$
$\displaystyle=\left({b}^{b^{\prime}},{\left({\left({a}^{\left({\left({a^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}\right)^{\left({\left({b^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}}\right)}^{b^{\prime}}+{\left({b}^{\left({\left({a^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}+{\left({b^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}\right)}^{b^{\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime}},\left({a}^{a^{\prime}}\right)^{b^{\prime}}+{\left({\left({b}^{\left({a^{\prime}}^{a^{\prime\prime}}\right)}\right)}^{\left({\left({b^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}+{b}^{\left({\left({b^{\prime}}^{a^{\prime\prime}}\right)}^{b^{\prime\prime}}\right)}\right)}^{b^{\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime}},\left({a}^{a^{\prime}}\right)^{b^{\prime}}+{\left({\left({b}^{a^{\prime}}\right)}^{\left({b^{\prime}}^{a^{\prime\prime}}\right)}\right)}^{b^{\prime}}+{\left({b}^{\left({b^{\prime}}^{a^{\prime\prime}}\right)}\right)}^{b^{\prime}}\right)$
$\displaystyle=\left({b}^{b^{\prime}},\left({a}^{a^{\prime}}\right)^{b^{\prime}}+{\left({b}^{a^{\prime}}\right)}^{b^{\prime}}\right)$
$\displaystyle={(b,a)}^{(b^{\prime},a^{\prime})}$

which proves the second identity. Here we applied that
${\left({b}^{\left({b^{\prime}}^{a^{\prime\prime}}\right)}\right)}^{b^{\prime}}=0$,
which follows from (4.1), where we have $b^{\left({b^{\prime}}^{a}\right)}=0$,
for any $a\in A,$ in particular for $a=a^{\prime\prime}$ in our case, and the
fact that $0^{b^{\prime}}=0$ (3.1 (b)).

  3. (3)$\Rightarrow$(1):

The proof is the same as of the one in Theorem 3.2 and therefore we omit.

∎

######  Lemma 4.3.

Let $A\in\mathbf{Gr}^{\bullet}$ (resp. $A\in\mathbf{rGr}^{\bullet}$). An
action of $A$ on itself defined by $a\cdot
a^{\prime}=a+a^{\prime}-a,a^{\prime}\ast a={a^{\prime}}^{\triangleright
a}={a^{\prime}}^{a}$ and $a^{\prime}{\ast}^{\circ}a=a^{\circ
a^{\prime}}={a}^{a^{\prime}}-a$, for $a,a^{\prime}\in A$, is a derived action
in $\mathbf{Gr}^{\bullet}$ (resp. $\mathbf{rGr}^{\bullet}$).

###### Proof.

Easy but careful checking of the conditions given in Theorem 3.2 (resp.
Theorem 4.2). ∎

Note, that an action of $A$ on itself defined by $a\cdot
a^{\prime}=a+a^{\prime}-a,a^{\prime\triangleright a}=a^{\prime a}$ and
$a^{\circ a^{\prime}}=a^{a^{\prime}}$, for $a,a^{\prime}\in A,$ is not a
derived action in $\mathbf{Gr}^{\bullet}$ and therefore in
$\mathbf{rGr}^{\bullet}.$ It is obvious that conditions $\bm{(2_{A})}$ and
$\bm{(1_{B})}$ are not satisfied.

######  Corollary 4.4.

Let $A\in\mathbf{Gr}^{\bullet}$ (resp. $A\in\mathbf{rGr}^{\bullet}$) and let
$I\subset A$ be an ideal of $A.$ Then the action of $A$ on $I$ defined by
$a\cdot i=a+i-a,i^{\triangleright a}=i^{a}$ and $a^{\circ i}=a^{i}-a$, $i\in
I$, $a\in A$ is a derived action in $A\in\mathbf{Gr}^{\bullet}$ (resp. in
$A\in\mathbf{rGr}^{\bullet}$).

Lemma 4.3 and Corollary 4.4 give examples of derived actions in the categories
$\mathbf{Gr}^{\bullet}$ and $\mathbf{rGr}^{\bullet}$.

## References

  * [1] Borceux, F., Janelidze, G. Z. and Kelly, G. M., Internal object actions, _Comment. Math. Univ. Carolin._ 46(2), (2005), 235–255.
  * [2] Boyacı, Y., Casas, J. M., Datuashvili, T. and Önder Uslu, E., Actions in modified categories of interest with application to crossed modules, _Theor. Appl. Categ._ 30, (2015), 882–908.
  * [3] Casas, J. M., Datuashvili, T. and Ladra, M., Actors in categories of interest, (arxiv:math.ct/0702574v2).
  * [4] Casas, J. M., Datuashvili, T. and Ladra, M., Universal strict general actors and actors in categories of interest, _Appl. Categor. Struct._ 18, (2010), 85–114.
  * [5] Datuashvili, T., Central series for groups with action and Leibniz algebras, _Georgian Math. J._ 9(4), (2002), 671–682.
  * [6] Datuashvili, T., Witt’s theorem for groups with action and free Leibniz algebras, _Georgian Math. J._ 11(4), (2004), 691–712.
  * [7] Datuashvili, T., Categorical, homological, and homotopical properties of algebraic objects, _J. Math. Sci._ 225(3), (2017), 383–533.
  * [8] Kurosh, A. G., _Lectures in General Algebra (Translated from Russian)_ , volume 70, Pergamon Press, Oxford-Edinburgh-New York (1965).
  * [9] Loday, J.-L., Une version non commutative des algèbres de lie : les algèbres de leibniz, _Enseign. Math. (2)_ 39(3-4), (1993), 269–293.
  * [10] Loday, J.-L., Algebraic $K$-theory and the conjectural Leibniz $K$-theory, _$K$ -Theory_ 30(2), (2003), 105–127.
  * [11] Orzech, G., Obstruction theory in algebraic categories, I, _J. Pure. Appl. Algebra_ 2(4), (1972), 287–314.
  * [12] Orzech, G., Obstruction theory in algebraic categories, II, _J. Pure. Appl. Algebra_ 2(4), (1972), 315–340.
  * [13] Porter, T., Extensions, crossed modules and internal categories in categories of groups with operations, _P. Edinburgh Math. Soc._ 30(3), (1987), 373–381.


# SeaLLMs - Large Language Models for Southeast Asia

Xuan-Phi Nguyen, Wenxuan Zhang∗, Xin Li∗, Mahani Aljunied∗, Qingyu Tan, Liying
Cheng,
Guanzheng Chen, Yue Deng, Sen Yang, Chaoqun Liu, Hang Zhang, Lidong Bing

DAMO Academy, Alibaba Group
Hupan Lab, 310023, Hangzhou, China

https://github.com/DAMO-NLP-SG/SeaLLMs Equal contributions. Corresponding
author<EMAIL_ADDRESS>
###### Abstract

Despite the remarkable achievements of large language models (LLMs) in various
tasks, there remains a linguistic bias that favors high-resource languages,
such as English, often at the expense of low-resource and regional languages.
To address this imbalance, we introduce SeaLLMs, an innovative series of
language models that specifically focuses on Southeast Asian (SEA) languages.
SeaLLMs are built upon the Llama-2 model and further advanced through
continued pre-training with an extended vocabulary, specialized instruction
and alignment tuning to better capture the intricacies of regional languages.
This allows them to respect and reflect local cultural norms, customs,
stylistic preferences, and legal considerations. Our comprehensive evaluation
demonstrates that SeaLLM-13b models exhibit superior performance across a wide
spectrum of linguistic tasks and assistant-style instruction-following
capabilities relative to comparable open-source models. Moreover, they
outperform ChatGPT-3.5 in non-Latin languages, such as Thai, Khmer, Lao, and
Burmese, by large margins while remaining lightweight and cost-effective to
operate.

##  1 Introduction

The advent of large-scale language models (LLMs) has radically transformed the
field of natural language processing, demonstrating remarkable abilities in
text generation, comprehension, and decision-making tasks [5, 21, 22, 32, 33,
31, 11, 36, 3]. While the proficiencies of these models are extraordinary, the
majority of existing LLMs embody a linguistic hierarchy overwhelmingly
dominated by English [1, 13, 37]. This dominance undermines the multilingual
capability of such models, with particularly prejudicial outcomes for lower-
resource and regional languages, where data scarcity and tokenization
challenges lead to disproportionately poor model performance. This linguistic
disparity not only impedes access to state-of-the-art AI technologies for non-
English-speaking populations but also risks cultural homogenization and the
loss of linguistic diversity. While hyper-polyglot models exist [26, 18, 36],
they may pay high cost at high-resource language performance and lack
multilingual instruction-following abilities.

Recognizing the urgent need to democratize AI and empower linguistically
diverse regions, we introduce SeaLLMs, a suite of specialized language models
optimized for Southeast Asian languages111English (Eng), Chinese (Zho),
Indonesian (Ind), Vietnamese (Vie), Thai (Tha), Khmer (Khm), Lao, Malay (Msa),
Burmese (Mya) and Tagalog (Tgl). These languages, while rich and diverse,
often lack the extensive dataset support available for more widely spoken
languages, resulting in a stark performance gap in existing LLM applications.

Our SeaLLMs-13B models eclipse the performance of most available open-source
LLMs in a comprehensive array of tasks including world knowledge assessments,
language comprehension, and generative capabilities in SEA languages. Compared
to the Llama-2-13B model, SeaLLMs does not only preserve, but also demonstrate
enhanced performance in tasks involving existing languages, such as English,
that were part of the original model’s training set.

Furthermore, our SeaLLM-Chat models emerge as formidable contenders to the
widely reputed ChatGPT [21], rivaling it as an AI assistant across multiple
interactive challenges, as corroborated by evaluations conducted with GPT-4 as
a judge [38]. This is despite the fact that GPT-4 evaluator favors ChatGPT
more often than humans do [12]. Notably, as shown Figure 1, our SeaLLM-13B
chat model outperforms ChatGPT-3.5 by large margins in less-represented
languages such as Khmer, Lao or Burmese.

Figure 2 illustrates the four-stage training process of SeaLLMs. In the first
stage, detailed in Section 2.3, we conduct continuous pre-training from the
foundational Llama-2 model [33] with an extended vocabulary tailored for SEA
languages. Next, we fine-tune the model in a novel hybrid paradigm with a
mixture of multilingual pre-training data and English-dominant instruction
fine-tuning data (Section 3.2). The following stage (Section 3.3) subsequently
fine-tunes the model on a balanced and custom-built multilingual SFT dataset.
Finally, we conduct self-preferencing alignment optimization using the SeaLLM
model itself, without relying on human annotators or more powerful LLMs [22].

Figure 1: Sea-bench (Section 4.2) scores as evaluated by GPT-4 [38] for
different models.Each radar chart compares scores as averaged across 5
categories (left) and 9 languages (right). Detailed breakdown by each category
and language is given in Figure 7 in the Appendix. Llama-2ContinualPre-
trainingPre-train & SFThybridSFTSelf-PreferencingOptimization Figure 2:
Complete Training Process of SeaLLMs. It begins with continual pre-training
Llama-2 with more data of regional languages. Then the models undergo
specialized fine-tuning process with multilingual SFT data, before finally
being tuned with self-preferencing alignment.

##  2 Pre-training

###  2.1 Pre-training Data

The pre-training data comprises a heterogeneous assortment of documents
sourced from several publicly accessible repositories [29, 25, 6, 9].
Specifically, during the creation of the pre-training data, we include web-
based corpora such as Common Crawl222https://commoncrawl.org, journalistic
content such as CC-News, text corpora with expertly-curated knowledge such as
Wikipedia [9], and some scholarly publications. After collecting the data, we
employ the FastText language identifier [4] to retain the documents for the
major languages in Southeast Asia, namely Thai, Vietnamese, Indonesian,
Chinese, Khmer, Lao, Malay, Burmese, and Tagalog, and discard the remaining
ones.

Subsequent stages of data refinement involve the deployment of a bespoke
pipeline configured with multiple modules dedicated to data cleansing and
content filtration. These modules are meticulously engineered to filter
content that is deemed harmful or otherwise inappropriate, thereby ensuring
that the integrity of the dataset is upheld.

Figure 3 (left) illustrates the composition of the pre-training data for
SeaLLMs. Even though the amount of available open-source English unlabeled
data is virtually unlimited in this scenario, we use only the highest quality
English data, sourced from a subset of RedPajama [6], in a more balanced blend
with the data from the SEA languages. The rationale for this is that most of
the English and Western-centric knowledge has already been learnt during
Llama-2 pre-training, and the English data used for SeaLLMs are mainly to
preserve such knowledge. This is vastly different from other polyglot models
[26, 36] which were pre-trained from scratch with data distributions that are
highly skewed towards English and high-resource languages.

The pre-training corpus is also heuristically categorized into high-quality
and low-quality data groups. High-quality data comes from sources that
comprise clean, regular, and formal texts, such as Wikipedia [9]. Meanwhile,
the low-quality data group includes sources with less formal and dirtier
content, such as web-based mC4 [25], and thus is an order of magnitude larger
than the high-quality data group. As we intend to train the model to lean more
towards high-quality data distribution, we adopt a novel pre-training data
streaming process that makes use of this data categorization, as explained
later in Section 2.3.

Figure 3: Language composition of Pre-training and SFT data for SeaLLMs. The
pre-training data has more balanced language ratios. However, the SFT data is
dominantly English because of the abundance of open-source English data and
the difficulty in acquiring SFT data in other languages.

###  2.2 Vocabulary Expansion

Table 1: Averaged compression ratios between the tokenized length of texts of each language produced by different tokenizers versus the baseline tokenized length of same-meaning English equivalents produced by ChatGPT tokenizer. For example, it costs 15.6x more tokens to encode the same English sentence in Khmer with ChatGPT tokenizer, while it only takes 2.67x more tokens with our tokenizer. Language | ChatGPT’s ratio | Llama’s ratio | SeaLLM’s ratio
---|---|---|---
Phase 1: Vie, Zho, Tha, Ind (+11K tokens)
Vie | 4.41 | 3.46 | 1.48
Zho | 2.80 | 2.36 | 1.40
Tha | 9.09 | 5.10 | 1.87
Ind | 2.00 | 2.09 | 1.36
Phase-2: Khm, Lao, Msa, Mya, Tgl (+5K tokens)
Khm | 15.56 | 12.14 | 2.67
Lao | 13.29 | 13.50 | 2.07
Msa | 2.07 | 2.16 | 1.50
Mya | 17.11 | 9.85 | 1.93
Tgl | 2.28 | 2.22 | 1.91
Eng | 1.00 (baseline) | 1.19 | 1.19

Table 1 describes how expensive it is to process an under-represented non-
Latin language. For example, encoding a single sentence in Thai requires
roughly 4.3 times more tokens than its English equivalent. The reason for this
is that most English/Latin-based language models employ a byte pair encoding
(BPE) tokenizer [27] that often inefficiently segments texts from non-European
and non-Latin scripts into disproportionately lengthy byte sequences. These
sequences inadequately represent the underlying semantic content, resulting in
diminished model performance [20]. Such inefficiency hampers the model’s
ability to handle tasks that depend on processing extended contexts, such as
reading comprehension, due to limitations imposed by maximum context lengths.

To that end, we propose a novel vocabulary expansion technique, as formally
described in Algorithm 1 in the Appendix. This technique involves recursively
merging whole-word and sub-word token pieces of a new language from a highly
multilingual target tokenizer, which in this case we use the NLLB tokenizer
[7], to the existing Llama-2 tokenizer. This new set of retrieved tokens are
then pruned to remove rarely appearing and low-quality tokens before being
added to the final SeaLLM tokenizer. This frequency-based pruning process
ensures the new language is sufficiently and efficiently encoded without
introducing tokens from other existing languages (e.g., English), which may
disrupt the learned knowledge during the Llama-2 pre-training stage. After
this extension process, we obtained 16,512 new tokens to represent Southeast
Asian languages.

Table 1 demonstrates the efficiency of the new vocabulary. The compression
ratio for Thai text has markedly improved from 4.29 to 1.57, signifying a
2.7-fold increase in the length of Thai text that can be encoded within the
same context constraints. At the same time, the compression of English text
has experienced a negligible reduction of 0.3%, thus maintaining its
tokenization effectiveness.

###  2.3 Pre-training Process

We organize our pre-training dataset based on the language of the content and
the quality of the data, as mentioned in Section 2.1. For each language, a
separate stream of data, which we denote as $F_{L_{i}}$, is established. When
we need a document for pre-training, we select a language, $L_{i}$, according
to a pre-defined probability, and then pull a document from the corresponding
stream $F_{L_{i}}$. This selection is done at each step when the model’s
parameters are updated.

We aggregate documents in possibly different languages into a single training
sequence. This helps the model learn a more language-agnostic representation
of texts and better cross-lingual knowledge transfer, while also maximizing
the training efficiency. This training sequence is designed to be as long as
the maximum context length. If the last document of the sequence is truncated
with a sufficiently long leftover, we move the cursor of the leftover to the
next new line to transform it into an uncorrupted document and then roll it
over to the next batch.

Our pre-training approach consists of two distinct stages. In the initial
stage, each language-specific stream $F_{L_{i}}$ feeds the model with
exclusively high-quality documents. This strategy helps guide the initial
learning process toward a preference for high-quality information. Once we
have used up all the high-quality data, the streams begin to supply a blend of
both high- and low-quality documents. In other words, streams for low-resource
languages may exhaust the high-quality data and begin with the high and low
mixture while other high-resource streams are still processing high-quality
data.

During the second phase, we continue to pre-train the model for a few thousand
more steps, but we revert to using only high-quality documents. This is
designed to readjust the model’s learning focus back towards the high-quality
data, improving the model’s performance.

##  3 Supervised Fine-tuning (SFT)

###  3.1 Supervised Fine-tuning Data

Our supervised finetuning (SFT) data consists of many categories, including
text understanding and processing, math and logical reasoning, user-centric
instruction-following, and natural dialog data. As most public and open-source
SFT data are English-only [16, 15, 19, 14], various techniques were
implemented to enhance the multilingual aspect of the model. These include
sourcing natural data from local websites in natural settings, selectively
translating from English data, employing self-instruction, and using advanced
prompting techniques [35, 17, 20]. As those synthetically generated data may
remain incorrect or low-quality, native speakers were then engaged to further
verify, filter, and edit such synthetic responses to finalize the SFT dataset.
We find that engaging the annotators to verify and modify model-generated
responses is more efficient than having them write responses from scratch.

Safety-related data also played a crucial role in fine-tuning SeaLLMs. This
included both universally applicable general safety practices and specific
safety protocols and norms relevant to different countries. We manually
collected and prepared country-relevant safety data, which covered a broad
range of culturally and legally sensitive topics in each of these countries.
This was necessary as such topics are often overlooked or may even conflict
with open-source English-centric safety data [8]. These measures ensured that
SeaLLMs gained a comprehensive understanding of safety practices across
different regions, enabling it to provide reliable responses in a multitude of
SEA languages.

Figure 3 (right) displays the composition of supervised fine-tuning data for
SeaLLMs. As can be seen, the significant majority of the data (96.7%) is
English and open-source, while the sizes of the SEA languages are minuscule.
As a result, a trivial implementation of supervised fine-tuning [23, 30] would
have overshadowed the multilingual dataset for SEA languages and caused the
model only to be able to respond in English. To address this issue, we propose
a pre-training and SFT hybrid fine-tuning stage as explained in the next
section Section 3.2.

###  3.2 Pre-training and Supervised Fine-tuning Hybrid

It is common practice to conduct supervised fine-tuning on instruction and
conversational datasets to enable a pre-trained model to obtain instruction-
following abilities [23, 33, 11, 22]. For the development of SeaLLMs, however,
we propose incorporating an additional step prior to complete fine-tuning,
namely Pre-train & SFT Hybrid. In this step, the model is further trained on a
combination of three types of data: the pre-training corpus, labeled data from
traditional NLP tasks, and significant quantities of open-source instruction-
following data. The rationale for this approach is that the open-source
English instruction data is available in dominantly larger quantities compared
to our SEA-language instruction data. This abundance can overshadow the
smaller SEA datasets during the SFT process. Thus, we reserve the majority of
English SFT data for this stage, while leaving a relatively small and balanced
amount of it to the next supervised fine-tuning process (Section 3.3).

During this intermediate, or hybrid, training stage, the model processes
batches that mix these different data types into singular sequences. In other
words, sequences may consist of both general pre-training content and
instruction-following examples. This blending differs from the initial pre-
training stage in that we avoid truncating unsupervised texts to preserve
their integrity. Additionally, we mask the source side of the instruction or
supervised data to prevent the model from overfitting to the training examples
and to reduce the risk of it simply memorizing the input data instead of
learning the more generalized ability to follow instructions.

###  3.3 Supervised Fine-tuning Process

Similar to Llama-2 [33], we initially conduct supervised fine-tuning by
compiling instructions from a variety of sources explained in Section 3.1,
combining them at random into a single, consolidated sequence. This technique
is designed to optimize the efficiency of the fine-tuning process. To further
enhance the usefulness and safety, we systematically integrate extensive and
carefully designed system prompts333e.g., “You are a helpful, respectful and
honest assistant. Your answers should not include any harmful, unethical,
racist, sexist, toxic, dangerous, or illegal content. Please ensure that your
responses are socially unbiased and positive in nature. If you don’t know the
answer to a question, please don’t share false information.” into every
instruction sample. Our initial research found that incorporating such
detailed, comprehensive and aligned system prompts is effective in
establishing a safe and useful prior within the model, which contributes to
the overall improvement of the model’s performance, as well as to increase its
adherence to safety protocols.

To enhance the multi-turn conversation capability, in the later stage of fine-
tuning, we further artificially create multi-turn conversations by randomly
joining several single-turn instructions together. Our preliminary experiments
suggest that this synthetic tuning approach significantly improves the models’
conversational capability.

###  3.4 Self-Preferencing Optimization

Alignment from human feedback preference has been key to the success of many
AI-assistant language models [28, 33, 24, 23]. To save the cost of human
preference annotation work, some has sought to use powerful LLMs like GPT-4
[22] to play as a preference data generator [34]. However, that may not even
be feasible for low-resource non-Latin languages because of the unfavorable
tokenization of ChatGPT as explained in Section 2.2. In other words, even
short prompts would exceed their context-length and the API-call costs would
explode by up to 17 times.

Therefore, we use our own SeaLLM SFT models to generate preference data using
a proprietary prompting strategy, which we later use to employ direct
preference optimization [24] to significantly improve the model abilities as
an AI agent. As such, unlike other works [19, 34], our models are free from
relying on powerful close-sourced models like GPT-4 to improve the performance
in low-resource languages.

##  4 Evaluation

###  4.1 Model Variants

We trained multiple variants of SeaLLMs, as specified in the following.

  * •

SeaLLM-7b: A 7B-parameter model trained to support all 10 languages. It is
more lightweight and fast.

  * •

SeaLLM-13b-5L: A 13B-parameter model trained in the first phase to support
English (Eng), Vietnamese (Vie), Indonesian (Ind), Thai (Tha) and Chinese
(Zho). This model has the best capability in these languages, with a
performance in Thai that rivals ChatGPT-3.5..

  * •

SeaLLM-13b-10L or SeaLLM-13b for short: A 13B-parameter model continually
trained from SeaLLM-13b-5L to support the remaining 5 official languages:
Khmer (Khm), Lao (Lao), Malay (Msa), Burmese (Mya) and Tagalog (Tgl). The pre-
training data for these languages is order of magnitude smaller than the other
corpora. This model outperforms ChatGPT-3.5 in most non-Latin SEA languages
(Khm, Lao, Mya and Tha) by large margins.

###  4.2 Sea-bench Peer Comparison

While there are popular benchmarks to evaluate LLMs as a helpful assistant,
such as MT-bench [38], they are only English-based and apparently not suitable
to evaluate performances in low-resource languages. Due to such a lack of
multilingual benchmarks for assistant-style models, we engaged native
linguists to build a multilingual test set with instructions that cover SEA
languages, called Sea-bench. The linguists sourced such data by translating
open-source English test sets, collecting real user questions from local
forums and websites, collecting real math and reasoning questions from
reputable sources, as well as writing test instructions themselves.

Our Sea-Bench consists of diverse categories of instructions to evaluate the
models, as follow:

  * •

Task-solving: This type of data comprises various text understanding and
processing tasks that test the ability of the language model to perform
certain NLP tasks such as summarization, translation, etc.

  * •

Math-reasoning: This includes math problems and logical reasoning tasks.

  * •

General-instruction data: This type of data consists of general user-centric
instructions, which evaluate the model’s ability in general knowledge and
writing. Examples for this type can be requests for recommendations, such as
"Suggest three popular books," or instructions that require the model to
generate creative outputs, like "Write a short story about a dragon."

  * •

NaturalQA: This consists of queries posted by real users, often in local
forums, about local contexts or scenarios. The aim is to test the model’s
capacity to understand and respond coherently to colloquial language, idioms,
and locally contextual references.

  * •

Safety: This includes both general safety and local context-related safety
instructions. The instructions could involve testing the model’s understanding
of safe practices, its ability to advise on safety rules, or its capacity to
respond to safety-related queries. While most general safety questions are
translated from open sources, other local context-related safety instructions
are written by linguists of each language. Safety data only covers Vietnamese,
Indonesian, and Thai.

As inspired by MT-bench [38], we evaluate and compare SeaLLMs with well-known
and state-of-the-art models using GPT-4 as a judge in a score-based grading
metrics and a peer comparison (or pairwise comparison) manner.

Figure 1 compares our SeaLLM-13b chat model with Qwen-14b [3], PolyLM-13b [36]
and the widely reputed ChatGPT-3.5 [21]. As shown in the figure, as aggregated
by task category (left radar chart), our SeaLLM-13b model performs on-par or
surpasses ChatGPT-3.5 across many linguistic and writing tasks. This is
despite the fact that GPT-4 evaluator favors ChatGPT more often than humans do
[12]. Our model do perform worse in terms of math reasoning, probably due to
the lack of math or code text sources, such as StackExchange or GitHub, in the
pre-training data of both SeaLLM and originally Llama-2 [33].

Meanwhile, the right radar chart of Figure 1 compares instruction-following
capabilities of models in the angle of different SEA languages. As shown,
SeaLLM-13b outperforms ChatGPT-3.5 by large margins in most non-Latin
languages, such as Burmese (Mya), Lao, Khmer and Thai. In combination with the
fact that SeaLLM can encode these languages with up to 9 times fewer tokens,
our models are not only superior but also cheaper to operate in these
languages than ChatGPT. This helps democratize the benefits of large language
models to under-represented and potentially developing communities.

On the other hand, Figure 4 evaluates Sea-bench as peer comparison between
SeaLLM-13b and ChatGPT-3.5. As it can be seen, despite potentially large model
size difference, SeaLLM-13b is equal or better than ChatGPT for 40% of the
times for Latin-based languages (Eng, Vie, Ind, Msa). In contrast, for non-
Latin languages, our SeaLLM-13b surpasses ChatGPT by up to 90%. The results
for peer comparison also appear to be consistent with score-based grading
results as shown in Figure 1.

(a) By language, averaged over 5 task categories

(b) By category, averaged over 9 languages

Figure 4: Sea-bench peer comparison between SeaLLM-13b v.s. ChatGPT as rated
by GPT-4 for different models across 9 languages and 5 categories.

###  4.3 World Knowledge

In this section, we evaluate our models and reputable chat baselines [33, 36,
21] in terms of world knowledge. For knowledge across languages, we use the
M3Exam benchmark [37], which consists of real questions from human exam papers
with various degrees of difficulty, ranging from primary school to high school
examinations. We evaluate M3Exam with 3-shot native-instruction prompts across
English, Chinese, Vietnamese, Indonesian and Thai. We also evaluate our models
with the well-known English-centric MMLU benchmark [10].

Table 2: Multilingual world knowledge accuracy evaluation across multiple languages and various models of different sizes. Model | M3Exam | MMLU
---|---|---
Eng | Zho | Vie | Ind | Tha | Eng
ChatGPT-3.5 | 75.46 | 60.20 | 58.64 | 49.27 | 37.41 | 70.00
Llama-2-7b | 49.58 | 37.58 | 29.82 | 28.93 | 19.89 | 45.62
Llama-2-13b | 61.17 | 43.29 | 39.97 | 35.50 | 23.74 | 53.50
Polylm-13b | 32.23 | 29.26 | 29.01 | 25.36 | 18.08 | 22.94
SeaLLM-7b | 54.89 | 39.30 | 38.74 | 32.95 | 25.09 | 47.16
SeaLLM-13b-5L | 63.20 | 45.13 | 49.13 | 40.04 | 36.85 | 55.23
SeaLLM-13b-10L | 62.69 | 44.50 | 46.45 | 39.28 | 36.39 | 52.68

Table 2 details the evaluations of world knowledge across multiple languages
and models of different sizes. Our SeaLLM-13b-5L model, which supports the 5
languages as specified in Section 4.1, performs the best across the
13B-parameter models. Meanwhile, the SeaLLM-13b-10L model shows slight
degradations as trade-offs for support for the remaining low-resource
languages.

###  4.4 Reading Comprehension

We also evaluate the reading comprehension capability of our SeaLLMs in
different SEA languages. Specifically, we utilize the evaluation sets from
XQuAD [2] for English, Chinese, Vietnamese, and Thai, and the evaluation set
from IndoQA444https://huggingface.co/datasets/jakartaresearch/indoqa for
Indonesian. We compare our SeaLLMs with vanilla Llama-2 of the same size under
a 1-shot setting and list the results of ChatGPT-3.5 as well for reference. As
shown in Table 3, SeaLLM models exhibit better performance for Vietnamese and
especially Thai. Meanwhile, ChatGPT’s scores may not reflect its true ability
for this task as it fails to comply with the shot format and instruction and
produces verbose words that decrease the F1 score.

Table 3: Reading comprehension F1 scores of different models across multiple languages. XQuAD [2] is used for Eng, Zho, Vie, and Thai, while IndoQA is used for Ind. Model | Eng | Zho | Vie | Ind | Tha
---|---|---|---|---|---
ChatGPT-3.5* | 69.18 | 69.66 | 59.25 | 52.82 | 40.75
Llama-2-7b | 80.28 | 71.92 | 64.23 | 50.86 | 26.18
SeaLLM-7b | 65.79 | 64.98 | 66.43 | 44.72 | 44.43
Llama-2-13b | 80.46 | 70.54 | 62.87 | 63.05 | 25.73
SeaLLM-13b-10L | 74.10 | 73.24 | 73.83 | 56.82 | 59.50

###  4.5 Machine Translation

To benchmark the machine translation performance of our SeaLLMs, we retrieve
the test sets from Flores-200 [7] as the evaluation data. In the experiments,
we adopt two evaluation protocols: (1) translation between English and SEA
languages and (2) translation between SEA languages directly. We report 4-shot
chrF++ scores for both SeaLLMs and the compared methods in Figure 5 and Figure
6. As can be seen, in terms of En$\leftrightarrow$X translation, our
SeaLLM-13B consistently surpasses the performance of LLaMA-2-13B with
reasonable margins. Besides, SeaLLM-13B exhibits clear superiority over
ChatGPT-3.5 in low-resource languages, such as Lao and Khmer, while
maintaining comparable performance with ChatGPT-3.5 in most high-resource
languages (e.g., Vietnamese and Indonesian). For direct translation between
SEA languages, our SeaLLM-13B still achieves higher chrF++ scores than
ChatGPT-3.5 in most cases, especially when the translation pairs involve low-
resource languages. Overall, compared to ChatGPT-3.5, our SeaLLM-13B
demonstrates remarkable capability in the translation from/to low-resource
languages and suffers very few or even no performance drops in the remaining
use cases. We believe our SeaLLMs will play a key role in facilitating
communication and cultural exchange across communities in Southeast Asia.

Figure 5: Translation chrF++ scores of various models for both SEA languages
to English and English to SEA languages directions. Figure 6: Direct
translation between SEA languages. Scores are indicated as the different
between the respective chrF++ score of SeaLLM-13b minus that of ChatGPT-3.5.
Red colors suggests SeaLLM-13b is better, while blue colors indicates ChatGPT
is better.

##  5 Conclusion

In conclusion, our research presents a substantial advance in the development
of equitable and culturally aware AI with the creation of SeaLLMs, a
specialized suite of language models attuned to the linguistic and cultural
landscapes of Southeast Asia. Through rigorous pre-training enhancements and
culturally tailored fine-tuning processes, SeaLLMs have demonstrated
exceptional proficiency in language understanding and generation tasks,
challenging the performance of dominant players such as ChatGPT-3.5,
particularly in SEA languages. The models’ attunement to local norms and legal
stipulations—validated by human evaluations—establishes SeaLLMs as not only a
technical breakthrough but a socially responsive innovation, poised to
democratize access to high-quality AI language tools across linguistically
diverse regions. This work lays a foundation for further research into
language models that respect and uphold the rich tapestry of human languages
and cultures, ultimately driving the AI community towards a more inclusive
future.

## References

  * Ahuja et al. [2023] Kabir Ahuja, Rishav Hada, Millicent Ochieng, Prachi Jain, Harshita Diddee, Samuel Maina, Tanuja Ganu, Sameer Segal, Maxamed Axmed, Kalika Bali, and Sunayana Sitaram.  MEGA: multilingual evaluation of generative AI.  _CoRR_ , abs/2303.12528, 2023.
  * Artetxe et al. [2019] Mikel Artetxe, Sebastian Ruder, and Dani Yogatama.  On the cross-lingual transferability of monolingual representations.  _arXiv preprint arXiv:1910.11856_ , 2019.
  * Bai et al. [2023] Jinze Bai, Shuai Bai, Yunfei Chu, Zeyu Cui, Kai Dang, Xiaodong Deng, Yang Fan, Wenbin Ge, Yu Han, Fei Huang, et al.  Qwen technical report.  _arXiv preprint arXiv:2309.16609_ , 2023.
  * Bojanowski et al. [2017] Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov.  Enriching word vectors with subword information.  _Transactions of the Association for Computational Linguistics_ , 5:135–146, 2017.  ISSN 2307-387X.
  * Brown et al. [2020] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901, 2020.
  * Computer [2023] Together Computer.  Redpajama: An open source recipe to reproduce llama training dataset, 2023.  URL https://github.com/togethercomputer/RedPajama-Data.
  * Costa-jussà et al. [2022] Marta R Costa-jussà, James Cross, Onur Çelebi, Maha Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe Kalbassi, Janice Lam, Daniel Licht, Jean Maillard, et al.  No language left behind: Scaling human-centered machine translation.  _arXiv preprint arXiv:2207.04672_ , 2022.
  * Deng et al. [2023] Yue Deng, Wenxuan Zhang, Sinno Jialin Pan, and Lidong Bing.  Multilingual jailbreak challenges in large language models.  _arXiv preprint arXiv:2310.06474_ , 2023.
  * [9] Wikimedia Foundation.  Wikimedia downloads.  URL https://dumps.wikimedia.org.
  * Hendrycks et al. [2021] Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt.  Measuring massive multitask language understanding.  _Proceedings of the International Conference on Learning Representations (ICLR)_ , 2021.
  * Jiang et al. [2023] Albert Q Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, et al.  Mistral 7b.  _arXiv preprint arXiv:2310.06825_ , 2023.
  * Koo et al. [2023] Ryan Koo, Minhwa Lee, Vipul Raheja, Jong Inn Park, Zae Myung Kim, and Dongyeop Kang.  Benchmarking cognitive biases in large language models as evaluators.  September 2023.
  * Lai et al. [2023] Viet Dac Lai, Nghia Trung Ngo, Amir Pouran Ben Veyseh, Hieu Man, Franck Dernoncourt, Trung Bui, and Thien Huu Nguyen.  Chatgpt beyond english: Towards a comprehensive evaluation of large language models in multilingual learning.  _CoRR_ , abs/2304.05613, 2023.
  * Lee et al. [2023] Ariel N. Lee, Cole J. Hunter, and Nataniel Ruiz.  Platypus: Quick, cheap, and powerful refinement of llms.  2023\.
  * Lian et al. [2023] Wing Lian, Bleys Goodson, Eugene Pentland, Austin Cook, Chanvichet Vong, and "Teknium".  Openorca: An open dataset of gpt augmented flan reasoning traces.  https://https://huggingface.co/Open-Orca/OpenOrca, 2023.
  * Longpre et al. [2023] Shayne Longpre, Le Hou, Tu Vu, Albert Webson, Hyung Won Chung, Yi Tay, Denny Zhou, Quoc V. Le, Barret Zoph, Jason Wei, and Adam Roberts.  The flan collection: Designing data and methods for effective instruction tuning, 2023.
  * Madaan et al. [2023] Aman Madaan, Niket Tandon, Prakhar Gupta, Skyler Hallinan, Luyu Gao, Sarah Wiegreffe, Uri Alon, Nouha Dziri, Shrimai Prabhumoye, Yiming Yang, et al.  Self-refine: Iterative refinement with self-feedback.  _arXiv preprint arXiv:2303.17651_ , 2023.
  * Muennighoff et al. [2022] Niklas Muennighoff, Thomas Wang, Lintang Sutawika, Adam Roberts, Stella Biderman, Teven Le Scao, M Saiful Bari, Sheng Shen, Zheng-Xin Yong, Hailey Schoelkopf, et al.  Crosslingual generalization through multitask finetuning.  _arXiv preprint arXiv:2211.01786_ , 2022.
  * Mukherjee et al. [2023] Subhabrata Mukherjee, Arindam Mitra, Ganesh Jawahar, Sahaj Agarwal, Hamid Palangi, and Ahmed Awadallah.  Orca: Progressive learning from complex explanation traces of gpt-4, 2023.
  * Nguyen et al. [2023] Xuan-Phi Nguyen, Sharifah Mahani Aljunied, Shafiq Joty, and Lidong Bing.  Democratizing llms for low-resource languages by leveraging their english dominant abilities with linguistically-diverse prompts.  _arXiv preprint arXiv:2306.11372_ , 2023.
  * OpenAI [2023a] OpenAI.  Chatgpt (june 2023 version, 2023a.
  * OpenAI [2023b] OpenAI.  Gpt-4 technical report.  _arXiv preprint_ , 2023b.
  * Ouyang et al. [2022] Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al.  Training language models to follow instructions with human feedback.  _Advances in Neural Information Processing Systems_ , 35:27730–27744, 2022.
  * Rafailov et al. [2023] Rafael Rafailov, Archit Sharma, Eric Mitchell, Stefano Ermon, Christopher D Manning, and Chelsea Finn.  Direct preference optimization: Your language model is secretly a reward model.  _arXiv preprint arXiv:2305.18290_ , 2023.
  * Raffel et al. [2019] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _arXiv e-prints_ , 2019.
  * Scao et al. [2022] Teven Le Scao, Angela Fan, Christopher Akiki, Ellie Pavlick, Suzana Ilić, Daniel Hesslow, Roman Castagné, Alexandra Sasha Luccioni, François Yvon, Matthias Gallé, et al.  Bloom: A 176b-parameter open-access multilingual language model.  _arXiv preprint arXiv:2211.05100_ , 2022.
  * Sennrich et al. [2016] Rico Sennrich, Barry Haddow, and Alexandra Birch.  Neural machine translation of rare words with subword units.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1715–1725. Association for Computational Linguistics, 2016.  doi: 10.18653/v1/P16-1162.  URL http://www.aclweb.org/anthology/P16-1162.
  * Stiennon et al. [2020] Nisan Stiennon, Long Ouyang, Jeffrey Wu, Daniel Ziegler, Ryan Lowe, Chelsea Voss, Alec Radford, Dario Amodei, and Paul F Christiano.  Learning to summarize with human feedback.  _Advances in Neural Information Processing Systems_ , 33:3008–3021, 2020.
  * Suárez et al. [2019] Pedro Javier Ortiz Suárez, Benoît Sagot, and Laurent Romary.  Asynchronous pipeline for processing huge corpora on medium to low resource infrastructures.  In _7th Workshop on the Challenges in the Management of Large Corpora (CMLC-7)_. Leibniz-Institut für Deutsche Sprache, 2019.
  * Taori et al. [2023] Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto.  Stanford alpaca: An instruction-following llama model.  https://github.com/tatsu-lab/stanford_alpaca, 2023.
  * Thoppilan et al. [2022] Romal Thoppilan, Daniel De Freitas, Jamie Hall, Noam Shazeer, Apoorv Kulshreshtha, Heng-Tze Cheng, Alicia Jin, Taylor Bos, Leslie Baker, Yu Du, et al.  Lamda: Language models for dialog applications.  _arXiv preprint arXiv:2201.08239_ , 2022.
  * Touvron et al. [2023a] Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, Aurelien Rodriguez, Armand Joulin, Edouard Grave, and Guillaume Lample.  Llama: Open and efficient foundation language models.  _arXiv preprint arXiv:2302.13971_ , 2023a.
  * Touvron et al. [2023b] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_ , 2023b.
  * Tunstall et al. [2023] Lewis Tunstall, Edward Beeching, Nathan Lambert, Nazneen Rajani, Kashif Rasul, Younes Belkada, Shengyi Huang, Leandro von Werra, Clémentine Fourrier, Nathan Habib, Nathan Sarrazin, Omar Sanseviero, Alexander M. Rush, and Thomas Wolf.  Zephyr: Direct distillation of lm alignment, 2023.
  * Wang et al. [2022] Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A Smith, Daniel Khashabi, and Hannaneh Hajishirzi.  Self-instruct: Aligning language model with self generated instructions.  _arXiv preprint arXiv:2212.10560_ , 2022.
  * Wei et al. [2023] Xiangpeng Wei, Haoran Wei, Huan Lin, Tianhao Li, Pei Zhang, Xingzhang Ren, Mei Li, Yu Wan, Zhiwei Cao, Binbin Xie, et al.  Polylm: An open source polyglot large language model.  _arXiv preprint arXiv:2307.06018_ , 2023.
  * Zhang et al. [2023] Wenxuan Zhang, Sharifah Mahani Aljunied, Chang Gao, Yew Ken Chia, and Lidong Bing.  M3exam: A multilingual, multimodal, multilevel benchmark for examining large language models.  _CoRR_ , abs/2306.05179, 2023.
  * Zheng et al. [2023] Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al.  Judging llm-as-a-judge with mt-bench and chatbot arena.  _arXiv preprint arXiv:2306.05685_ , 2023.

##  Appendix A Vocabulary Expansion

Algorithm 1 explains in details how we perform selective and recursive merger
of tokens from target NLLB vocabulary into the original Llama vocabulary to
enrich the linguistic coverage for new and low-resource languages.
Specifically, given a small seed unlabeled dataset of a given new language,
the algorithm first tokenizes a document with the current Llama tokenizer. The
resulting tokens are then exhaustively merged into longer tokens that are
supported by the target NLLB vocabulary. During this merger process, any
intermediate sub-word is also added to the Llama tokenizer as long as they
exist in the rich NLLB vocabulary.

The new set of collected tokens are then pruned to remove rarely appearing and
low-quality tokens before being added to the final SeaLLM tokenizer. This
frequency-based pruning process ensures the new language is sufficiently and
efficiently encoded without introducing tokens from other existing languages
(e.g., English), which may disrupt the learned knowledge during the Llama-2
pre-training stage.

Algorithm 1 Vocabulary Extension algorithm: $V_{i}$ is Llama vocabulary,
$V_{t}$ is target NLLB vocabulary, $D$ is unlabeled data and $m$ is minimum
frequency.

1:function ExhaustiveMerge($V_{i},V_{t},t_{V}$)

2: $T_{new}\leftarrow\text{empty set }\emptyset$

3: repeat

4: for each consecutive token pair $(\text{prev},\text{next})$ in $t_{V}$ do

5: $t_{merged}\leftarrow\langle\text{prev}\rangle\langle\text{next}\rangle$
$\triangleright$ Form a new token

6: if $t_{merged}$ exists in $V_{t}$ then

7: Replace $(\text{prev},\text{next})$ with $t_{merged}$ in $t_{V}$
$\triangleright$ Update $t_{V}$ with new token

8: $T_{new}\leftarrow T_{new}\cup t_{merged}$

9: break

10: until no new token added to $T_{new}$

11: return $T_{new}$

12:function VocabExtend($V_{i},V_{t},D,m$)

13: $V\leftarrow V_{i}$

14: $F\leftarrow\text{empty set }\emptyset$

15: $T\leftarrow\text{empty set }\emptyset$

16: for document $d$ in $D$ do

17: $t_{V}\leftarrow\text{tokenize}(V,d)$ $\triangleright$ tokenize the
document

18: $T_{new}\leftarrow\textsc{ExhaustiveMerge}(V_{i},V_{t},t_{V})$
$\triangleright$ obtain new words from $V_{t}$ based on $d$

19: $V\leftarrow V\cup T_{new}$ $\triangleright$ update $V$ with new words
$T_{new}$

20: $T\leftarrow T\cup T_{new}$

21: $F\leftarrow$ Update frequencies of $T_{new}$ to $F$ $\triangleright$
update appearance frequencies of $T_{new}$

22: $T\leftarrow$ Prune $t_{i}\in T$ with corresponding $f_{t}\in F$ where
$f_{t}<m$ $\triangleright$ Remove rare words

23: $V_{final}\leftarrow V_{i}\cup T$

24: return $V_{final}$

##  Appendix B Sea-bench Evaluation Details

Figure 7 breaks down the GPT-4 rated Sea-bench score-based evaluations of
SeaLLM-13b and other baselines by both language and task category. As shown,
our SeaLLM-13b model far exceeds ChatGPT-3.5 in most non-Latin languages, such
as Burmese (Mya), Lao and Khmer, though it trails behind this formidable
competitor in Latin-based languages, mostly in math reasoning skills.

Figure 7: Sea-bench scores as evaluated by GPT-4 for different models across 9
languages and 5 categories.