Inclines are additively idempotent semirings in which products are less than (or) equal to either factor. Necessary and sufficient conditions for an element in an incline to be regular are obtained. It is proved that every regular incline is a distributive lattice. The existence of the Moore-Penrose inverse of an element in an incline with involution is discussed. Characterizations of the set of all generalized inverses are presented as a generalization and development of regular elements in a ^{∗}-regular ring.

The notion of inclines and their applications are described comprehensively in Cao et al. [

In [

In this paper, we exhibit that Green’s equivalence relations on a pair of elements in an incline reduce to the equality of elements. This leads to the characterization of regular element in an incline that is, an element in an incline is regular if and only if it is idempotent and structure of set of all

Green’s equivalence relation reduces to equality of elements. We conclude that the proofs are purely based on incline property without using star cancellation law as in the work of Hartwig [

In this section, we give some definitions and notations.

An incline is a nonempty set

An incline

For

For

Thus

For

Throughout let

For any two elements

In this section, equivalent conditions for regularity of an element in an incline are obtained and it is proved that a regular commutative incline is a distributive lattice. The equality of right (left) ideals of a pair of elements in a regular incline reduces to the equality of elements. This leads to the invariance of the product

Just for sake of completeness we will introduce

An element

For

An incline

The Fuzzy algebra

Let

Let

If

Therefore

Similarly, from

For

Let

Converse is trivial.

Let us consider the example

If

Let

It is well known that [

A commutative incline is a distributive lattice as (semiring) if and only if

DL is a distributive lattice. (DL is the set of all idempotent elements in an incline L.)

Let

Let

Conversely, if

Next we shall see some characterization of regular elements in an incline.

For

group inverse of

(i)

To prove the theorem it is enough to prove the following implications:

(ii)

(ii)

Lemma

Thus (iii) holds.

(iii)

Therefore by Definition

(iv)

(i)

Thus (v) holds.

(v)

Therefore

Thus (ii) holds.

(i)

Now if

Thus

In a similar manner we can show

Now consider,

Hence,

Thus

If

Let us illustrate the relation between various inverses associated with an element in an incline in the following.

Let

In this incline

Hence

Since

Hence

Let

If

Whenever two symmetric results are involved we shall prove the first leaving the second.

Let

Let

Since

Hence (i) holds.

Let

Conversely, let

Then,

Interchange

Let

Now,

Thus (iv) holds.

Let

Let

From the statement (ii), we have

Therefore

Now,

Thus (vi) holds.

For a, b in a regular incline one has the following:

Since

By Theorem

Therefore

On the other hand

We note that Corollary

Let us consider

Since

By Proposition

Hence

It is well known that [

Let

Let

Therefore

Pre- and postmultiplication by

By Property

Thus for each

On the other hand for any

From Definition

In this section, the existence of the Moore-Penrose inverse of an element in an incline with involution-

An involution-

An element

For

An element

In [^{T}^{T}

First we shall show that Green’s equivalence relation on an incline

For

(i)

Converse holds for elements in a regular incline or incline with unit.

(

Therefore,

(

The converse holds for

Let R be an incline with involution-T. For

a^{†} exists and equals

a^{T}a^{T} has a solution in R,

^{T} = a^{T} has a solution in R,

a is regular and a^{T}

a^{T}a,

Let

This can be proved along the same lines as that of (i)

This equivalence can be proved directly by verifying that

Let

Conversely, if

Now,

This can be proved in the same manner and hence omitted.

(vii)

It is well known that [

Let us consider the incline

Hence

Let R be an incline with involution-T. For any element

This can be proved along the same lines as that of Theorem

The main results in the present paper are the generalization of the available results shown in the reference for elements in a *-regular ring [

In [