#### Abstract

All unital four-dimensional lattice-ordered algebras with a -basis are constructed.

#### 1. Introduction

Lattice-ordered algebras (-algebras) with a distributive basis (-basis) were studied in [5] in an effort to obtain general structure theory for a class of -algebras that includes some well-known and important examples of -algebras, such as matrix and triangular matrix -algebras over a totally ordered field with entrywise lattice order, polynomial -algebra over a totally ordered field with coordinatewise lattice order, and so forth.

All unital two-dimensional and three-dimensional -algebras with a -basis have been constructed [6]. In this paper, we construct all unital four-dimensional -algebras with a -basis. We will first review some definitions and results for later use. The reader is referred to [1, 5, 6] for the general theory of lattice-ordered rings (-rings) and undefined terminologies in this paper. Throughout always denotes a totally ordered field.

Let be an -ring. The positive cone of is . A vector lattice over is an -group and a vector space over such that . An -algebra over is an algebra over , that is, an -ring and a vector lattice over . An -algebra is called unital if it has an identity element 1 and called -unital if . An element in an -ring is called basic if and, for any , implies that and are comparable, that is, either or . A set of elements is called disjoint if each and for any two different elements and . An element in an -ring is called an -element if implies that , and is called a -element if implies that . An -ring is called -reduced if for any , implies that .

Let be an -algebra over . A nonempty subset of is called a -basis of if is a disjoint subset of that spans as a vector space over and each element in is a -element [6, Definitionββ2.1]. Then for each , , where , , and are distinct, and if and only if each , . Thus in a -basis, each element is basic. Let be a unital finite-dimensional -algebra with a -basis. Then is -unital [1], and there exists a -basis such that , where are distinct.

In the following theorem, we record some results on unital -algebras with a -basis from [5, 6].

Theorem 1.1. Let be a unital finite-dimensional -algebra over with a -basis such that , where and are distinct. (1)Each is an idempotent -element and for . (2)Let be a basic element. Then there exists exactly one such that , and for any , . Similarly, there exists exactly one such that , and for any , . (3)For each element , if or , , then either is nilpotent or for some positive integer and . (4)Let . If either or is not nilpotent and , then is basic.

#### 2. The Classification of Unital Four-Dimensional β-Algebras with a π-Basis

We first list some -algebras that we are going to use for the classification of unital four-dimensional -algebras over with a -basis.(i)Let be a finite group, and let be the group algebra over . Define an element if each . Then is an -algebra over and is a -basis. In the following, always denotes the -algebra with the lattice order defined above.(ii)Let , and let be irreducible. We use to denote a fixed root of . Consider the extension field of . Then and is a basis of over as a vector space over . Now define if each , . Since , becomes an -algebra over with respect to the above lattice order, and is a -element. Thus is a -basis of the -algebra over . In this paper, and always denotes the -algebra defined above.(iii)For , let and be matrix algebra and upper triangular matrix algebra, respectively. Define a matrix positive if each of its entries is positive. Then and become -algebras. Let denote the matrix with entry equal to 1 and other entries equal to 0. Then is a -basis for over and is a -basis for over . In the following, and always denote the -algebras with the above lattice order.

All nonisomorphic unital four-dimensional -algebras with a -basis are listed in Theorem 2.1. Suppose that is a unital four-dimensional -algebra with a -basis such that 1 is a sum of some distinct elements in . In the proof of Theorem 2.1, we consider four different cases according to the number of elements in this sum, and in each case we consider two cases in which is -reduced and is not -reduced, respectively. We use to denote a direct sum of -rings.

Suppose that is a four-dimensional -algebra with a -basis such that . By Theorem 1.1, each is an idempotent -element, and for , , . Thus is a four-dimensional -algebra. This is (1) in Theorem 2.1.

In the following sections, we consider the remaining three cases where 1 is a sum of three, two, or one element from the -basis , respectively.

Theorem 2.1. Let be a unital four-dimensional -algebra over with a -basis. Then is isomorphic to one of following -algebras over : (1), (2), where is a group of order two, (3), where , (4), where , (5), (6), where is a group of order two, (7), where and is a group of order two, (8), where , (9), where is a group of order three, (10), where , (11), (12), where , (13), where , (14), where , , and , (15), where , and , (16), where , , and , (17), where , , and , (18), where , (19), where , (20), where and , (21), where is a cyclic group of order four, (22), where , (23), where is a group of order two and with and , (24), where is the Klein four group, (25), where and is a group of order two, (26), where , , (27), where and the product of any other two elements of is zero, (28), where , (29), where , , and the product of any two elements not both or is zero, (30), where the product of any two elements not both is zero, (31), where is a zero ring, (32), where , (33), where and with , (34), where is a group of order two, , and commutes with each element in , (35), where and commutes with each element in .

#### 3. 1 Is a Sum of Three Disjoint Basic Elements

Let be a unital four-dimensional -algebra with a -basis such that . Then by Theorem 1.1, each is an idempotent -element, , and for any , . By Theorem 1.1, we may assume that and . We consider two cases according to whether is -reduced or not.

##### 3.1. π΄ Is β-Reduced

If , then for or 3 by Theorem 1.1(2), and hence , which contradicts the fact that is -reduced. Thus and . Since is not nilpotent, by Theorem 1.1(4), is a basic element, so for some and . Since and , . If or , then , which is a contradiction. Thus for some . There are two cases now. (1)There exists such that . Let . Then is also a -basis with the following multiplication table: Thus , where is a group of order two. This is (2) in Theorem 2.1. (2)The polynomial has no root in . Then we have the following multiplication table for : Thus . This is (3) in Theorem 2.1.

##### 3.2. π΄ Is Not β-Reduced

In this case, must be nilpotent. If , then must be a positive linear combination of elements in . Since each of is not nilpotent, we must have for some . Then is nilpotent implying that , which is a contradiction. Thus we must have . There are two cases that we need to consider. (1). Then , and the multiplication table for is given below: Thus , where and . This is (4) in Theorem 2.1. (2). Without loss of generality, we may assume that and . Then we have the following multiplication table for elements in : Let be the upper triangular matrix algebra over with the entrywise lattice order. Then becomes an -algebra over with a -basis . The multiplication table for is given below: Since and have the same multiplication tables, we have . This is (5) in Theorem 2.1.

#### 4. 1 Is a Sum of Two Disjoint Basic Elements

Let be a unital -algebra with a -basis such that . By Theorem 1.1, for some or 2. Then implies that . Similarly . Thus for some . By a similar argument, , for some .

##### 4.1. π΄ Is β-Reduced

We may assume that and . Since contains no nonzero positive nilpotent elements, we also have and . Then implies that . Similarly, .

Since is -reduced, is not a nilpotent element, so is basic by Theorem 1.1. Thus for some and . Since and , . If , then , which is a contradiction. Thus or . We consider these two cases separately. (1). Suppose that . Since , , so , which is a contradiction. Thus , and hence since . Since , . Thus , and hence and . Now is not nilpotent implying that is a basic element, so for some and . Since and , . If or , then or , which are contradictions. Thus we must have . Then we have the following multiplication table for the elements of : Depending on or and or , we have the following cases: where is a group of order two. These are (6), (7), and (8) in Theorem 2.1. (2). Since with , is also a -basis for the -algebra over and . Let . Then since , so we have the following multiplication table for elements in : Now it is clear that if , then , where is a group of order three, and if , then , where is the -field extension of of order three. Those are (9) and (10) in Theorem 2.1.

##### 4.2. π΄ Is Not β-Reduced

Since is not -reduced, or is nilpotent. Let's assume that , and hence . Suppose that is nilpotent and . Then Let be the smallest positive integer such that . Then and so . Thus , and hence . Then , which implies that and . Then implies that . So is a -basis of over . Similarly if is nilpotent and , then is a -basis of over . Therefore we only need to consider the following cases.(i).(ii) is nilpotent, but . By the above argument, for some . Thus is also a -basis for over .(iii) and is not nilpotent.

Below, we consider each case in detail.

###### 4.2.1. π2=π2=0

From , we have , so , and hence . (1). Then implies that , , , and . Then implies that , so and . Thus implies that . Since , and . Let . Then is also a -basis of with the following multiplication table: It is clear now that , the matrix -algebra over . This is (11) in Theorem 2.1. (2). Then . From , we have , so . Thus , and hence , which implies that , so . Therefore, is a zero -ring. There are the following cases.(2a)If , , and , then , where is the zero -ring.(2b)If , , and , then , where . If , then the multiplication table for is given below: Clearly is isomorphic to , and hence there are the following cases depending on the products between , , and .(2c)If and , then with , , and .(2d)If and , then with , , and .(2e)If and , then with , , , and .(2f)If and , then with , , and .

Two more cases we need to consider are (i) , , and (ii) , , . But when we switch and , the case (i) becomes (2c), and when we switch and , and also switch and , the case (ii) becomes (2f). Thus every possible case when is covered. The above discussion has produced (12)β(17) in Theorem 2.1.

###### 4.2.2. π Is Nilpotent and π2β 0

As we have noticed before, in this case is a -basis for the -algebra over . Since implies that and implies that , , for some . If , then cannot be nilpotent, so we must have . Therefore , and with . This is (18) in Theorem 2.1.

###### 4.2.3. π2=0 and π Is Not Nilpotent

Since is not nilpotent, is basic, so for some and . Clearly or . Thus either or . In the first case, we have , so , which is a contradiction. Thus we must have . If , then , so , which implies that , a contradiction. Thus , and hence . Then . Therefore we have the following multiplication table for the elements of :

From the above table, it is clear that if , then with , where is a group of order two, and if , then with , where is a quadratic extension -field of . Those are (19) and (20) in Theorem 2.1.

Now we have constructed all lattice orders on such that 1 is a sum of two disjoint basic elements.

#### 5. 1 Is a Basic Element

In this section, has a -basis . We still first consider the -reduced case.

##### 5.1. π΄ Is β-Reduced

Since is not nilpotent, by Theorem 1.1, there exist positive integer and such that . We also assume that is the smallest positive integer such that . Similarly and , for some positive integers , , and . By Theorem 1.1 again, each is a basic element for .

Lemma 5.1. The subset of is a disjoint set of .

Proof. Let . Since is a basic element, for some and . Since , , so implies that . Now let , then by the above argument.

Since is four-dimensional, by Lemma 5.1. We claim that . Supposing that , then we may assume that is a -basis. By Theorem 1.1, is a basic element and , , , so for some . Thus , and hence , which is a contradiction. Hence , so or .

By the above analysis, we need to consider two cases: (i) one of the following: , , , is 4, and we may assume that ; (ii) .

###### 5.1.1. ππ=4 and Hence {1,π,π2,π3} Is a Disjoint Subset of Basic Elements of π΄

In this case is a -basis of over , and we have the following multiplication table for the elements in :

We have the following three cases to be considered.(1)If , then , where is a cyclic group of order four.(2)If the polynomial is irreducible, then , which is an -field extension of of order four.(3)The polynomial is not irreducible and .

Suppose that . Then we have , , , and . Thus and . If , then , and hence , which implies that , a contradiction. So we must have , and hence . Then and implies that and . We may assume that . It follows from that . Let . Then is also a -basis of over and the multiplication table for the elements in is given below:

Then , where is a group of order two and , , and . These are (21), (22), and (23) in Theorem 2.1.

###### 5.1.2. ππ=ππ=ππ=2

Then is a -basis and , , and . Since and is basic, or for some . In the first case, implies that , which is a contradiction. Thus for some . Then we may replace by to get a -basis .

Since and is basic, or for some . If , then , which is a contradiction. Thus , for some , and hence So and the multiplication table for the elements of is given below:

There are the following cases. (1) and . Replacing and by and , respectively, we have . Therefore , where is the Klein four group. (2) and (or and ). Replacing by , we may assume that . Therefore , where is the quadratic extension -field of and is a group of order two.(3) and . If , then , where is a group of order two. We omit the checking and leave it to the reader. If , then , where is a quadratic extension -field of and is a quadratic extension -field of . Those are (24), (25), and (26) in Theorem 2.1.

##### 5.2. π΄ Is Not β-Reduced

Let be a -basis of over . Then some elements of must be nilpotent. We first observe the following result.

Lemma 5.2. Let . If either or is nilpotent and the other one is not nilpotent, then is nilpotent.

Proof. Let be nilpotent and let be not. By Theorem 1.1, there exist a positive integer and such that . If , then implies that , which is a contradiction. Thus , so is basic by Theorem 1.1. If is not nilpotent, then for some and and is not nilpotent. By Theorem 1.1, there exist a positive integer , such that . Let be the smallest positive integer such that . Then , and hence implies that . Thus , which is a contradiction. Therefore is nilpotent.

By Lemma 5.2, cannot have exactly one nilpotent element. Suppose not, let be the unique nilpotent element. Since , cannot be a positive scalar multiple of , so cannot be a nilpotent element, which is a contradiction by Lemma 5.2. So there are following two cases.

###### 5.2.1. π, π, and π Are All Nilpotent
(1)Either , , or has nonzero square. We may assume that . Since , , where . Let be the smallest positive integer such that . Then implies , so . Thus .(1a) and . Since , . Also . Thus . Then we have , so and . Similarly , , and . Since , implies that . Similarly . Replacing and by and , respectively, we may assume that . Therefore we have the following multiplication table: Then , where and the product of any other two elements of , , is zero. This is (27) in Theorem 2.1. We want to point out that this case provides an example in which the product of two basic elements may not be basic. (1b) or . We may assume that , so . Then , and implies that , so .(i). Then for some . Thus , and hence we may replace by and by to get another -basis . Therefore with . This is (28) in Theorem 2.1.(ii). Then , so . Since for some , and implies that , so . Similarly . , so . Similarly, . Now replacing by , we obtain another -basis .If , then implies that . Then it follows from and that , so . Thus with . Hence with for some and for any two elements and in not both or . This is (29) in Theorem 2.1. If , then with for any two elements and in which are not both . This is (30) in Theorem 2.1.We provide some familiar examples for the above two cases. Consider matrix algebra over . Let , , and . Then and . Clearly is a four-dimensional -algebra over with coordinatewise order. If , then and is a four-dimensional -algebra over with . (2). (2a) for any two elements in .Then , where is zero ring. This is (31) in Theorem 2.1. (2b)At least one product of two different elements in is nonzero. Without loss of generality, we may assume that . Then for some , and we may replace by to form another -basis .If , then we have the following multiplication table: Thus with . This is (32) in Theorem 2.1. If , then for some and we have the following multiplication table: Thus with and for some . This is (33) in Theorem 2.1.
###### 5.2.2. Two Elements in {π,π,π} Are Nilpotent

We may assume that and are nilpotent elements. Then we must have for some . By Lemma 5.2, for some . If , then and that 1 and are not nilpotent implies that for some . Thus , which is a contradiction, so . Similarly, . Replacing by and then replacing by , we have another -basis with the following multiplication table:

If , then replacing by , we have that is a commutative -algebra, where is a group of order two and .

If , then is a commutative -algebra over , where is a quadratic extension -field of and . Those are (34) and (35) in Theorem 2.1.

This completes the proof of Theorem 2.1.

#### 6. An Application

For an -ring , its -radical - is defined as (see [1, Definition, page 45]). - is the sum of all of the nilpotent -ideals of [1, Theoremββ5]. It is clear that for an -algebra , - is an -ideal of .

A positive derivation on an -ring is a function such that for any , , , and . For an Archimedean -ring , if is a positive derivation on , then - [2, 3]. Although we believe this result is true for a unital -algebra with a -basis, we lack ability to prove it in general. In the following, we prove this result for unital four-dimensional -algebras with a -basis based on the classifications obtained in Theorem 2.1.

Theorem 6.1. Let be a unital four-dimensional -algebra with a -basis. If is a positive derivation on , then .

Proof. We notice first that since , for any and , [4]; that is, is actually a linear transformation of . Let , where are distinct elements from the -basis and . Since implies , . Then is positive implying that for each . Let be an element in the -basis, by Theorem 1.1, either is nilpotent or for some , , and . If is nilpotent, then is also nilpotent [4]. If , then and is positive also implying that is nilpotent. Thus for each element in the -basis, is nilpotent, so if is -reduced, then for each in the -basis. Therefore .
We only need to consider those cases in Theorem 2.1 in which is not -reduced. In cases (4)-(5), (12)β(20), and (27)β(35), the -radical consists of all nilpotent elements of the -algebra . From the argument in the previous paragraph, for each element in the -basis, is nilpotent, so -. Thus - for those cases.
The only case left is (11) in which -, but contains nonzero nilpotent elements and . We show that . Then together with , we have . Since , so is positive implying that . Multiplying this last equation by from the left, we have . Hence . Therefore .
This completes the proof.