### Article

Kyungpook Mathematical Journal 2017; 57(1): 99-108

**Published online** March 23, 2017

Copyright © Kyungpook Mathematical Journal.

### On the Construction of Polynomial β -algebras over a Field

Keum Sook So^{1}

Young Hee Kim^{2}

Department of Mathematics, Hallym University, Chuncheon 24252, Korea^{1}

Department of Mathematics, Chungbuk National University, Chongju 28644, Korea^{2}

**Received**: September 22, 2016; **Accepted**: November 10, 2016

### Abstract

In this paper we construct quadratic

**Keywords**: (linear, quadratic-linear, linear-quadratic) $eta$-algebra, (left, right)-zero semigroup, $BCK$-algebra

### 1. Introduction

Y. Imai and K. Iséki introduced two classes of abstract algebras: ^{*}-algebra, then (

In this paper we construct quadratic

### 2. Preliminaries

A

(i) (((

x *y ) * (x *z )) * (z *y ) = 0),(ii) ((

x * (x *y )) *y = 0),(iii) (

x *x = 0),(iv) (

x *y = 0,y *x = 0 ⇒x =y ).(v) (0 *

x = 0),

(0 −

(

### Example 2.1.([9])

Let

+ | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 1 | 2 | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

− | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 3 | 2 | 1 |

1 | 1 | 0 | 3 | 2 |

2 | 2 | 1 | 0 | 3 |

3 | 3 | 2 | 1 | 0 |

Then (

Given a ^{* }:= 0 −

### Theorem 2.2.([6])

We call such a

If we let

and

so that

### Proposition 2.3.([6])

A

An algebra (

If a

### 3. Constructions of Polynomial β -algebras

Let (

where

From (

If we assume that |

Case(i).

and

By (III), we obtain

Case (ii).

It follows that ^{2}. This shows that

which means that

It is a quadratic form (not a linear form) and so (

### Theorem 3.1

### Corollary 3.2

**Proof**

It follows immediately from Theorem 3.1 by letting

### Example 3.3

Let ^{2}−2

Let (

### Theorem 3.4

**Proof**

Let (

where

It follows that

Case (i).

It follows that 0 ⊖ ^{2 }= ^{2 }and hence

for all

Subcase (i-1).

and

By (III), we obtain

Subcase (i-2).

and

By (III), we obtain

Case (ii). ^{2},

Subcase (ii-1).

If ^{2 }− ^{2}) = ^{2 }− ^{2}). It follows that

for all

Since ^{2}−^{2}) does not form a ^{2 }and (

Subcase (ii-2).

It follows that

for all

Hence

Using formulas (

By formula (

By (III), we have

Subcase (ii-2-a).

Subcase (ii-2-b).

for all

### Problem

Construct a complete quadratic

### 4. Some Relations of Binary Operations in β -algebras

In this section, we discuss some relations of binary operations in

### Proposition 4.1

**Proof**

Assume that (

### Proposition 4.2

**Proof**

Since (

### Corollary 4.3

**Proof**

Straightforward.

### Proposition 4.4

**Proof**

Let (

### Proposition 4.5

**Proof**

Assume (

### Proposition 4.6

**Proof**

Assume (

### Theorem 4.7

**Proof**

Assume ^{2 }+ ^{2}. Then

It follows that ^{2 }=

i.e.,

Using Theorem 4.7 and Proposition 4.2, we obtain the following.

### Corollary 4.8

### Tables

+ | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 1 | 2 | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

− | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 3 | 2 | 1 |

1 | 1 | 0 | 3 | 2 |

2 | 2 | 1 | 0 | 3 |

3 | 3 | 2 | 1 | 0 |

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