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We introduce the concepts of set-valued homomorphism and strong set-valued homomorphism of a quantale which are the extended notions of congruence and complete congruence, respectively. The properties of generalized lower and upper approximations, constructed by a set-valued mapping, are discussed.

The concept of Rough set was introduced by Pawlak [

The majority of studies on rough sets for algebraic structures such as semigroups, groups, rings, and modules have been concentrated on a congruence relation. An equivalence relation is sometimes difficult to be obtained in real-world problems due to the vagueness and incompleteness of human knowledge. From this point of view, Davvaz [

In this section, we give some basic notions and results about quantales and rough set theory (see [

A

An element

A quantale

A quantale

A subset

In a quantale

Let

A subset

Let

An ideal of

An ideal

An ideal

A nonempty subset

A nonempty subset

Every ideal of

Let

Let

Let

Let

The pair

From the definition, the following theorems can be easily derived.

Let

If

Let

In this paper,

Let

(1) Suppose that

(2) Suppose that

(3) It is obvious that

(4) Since

(5) and (6) The proofs are similar to (1) and (2), respectively.

A set-valued mapping

(1) Let

(2) Let

Let

(1) Suppose that

(2) The proof is similar to (1).

(3) Suppose that

(4) Suppose that

Let

(1) Suppose that

(2) The proof is similar to (1).

Let

If

Since

If

Let

Suppose

Let

Suppose that

Let

Let

Let

Let

Let

Since

Since

Let

The proof is similar to Theorem

Let

Suppose that

Since

Suppose

Let

Suppose

Since

Suppose

A subset

The following corollary follows from Theorems

Let

From the above, we know that an ideal is a generalized rough ideal with respect to a strong set-valued homomorphism. The following example shows that the converse does not hold in general.

Let

Let

0 | 1 | ||
---|---|---|---|

0 | 0 | 1 | |

0 | 1 | ||

1 | 0 | 1 |

0′ | 1′ | |||
---|---|---|---|---|

0′ | 0′ | 1′ | ||

0′ | 1′ | |||

0′ | 1′ | |||

1′ | 0′ | 1′ |

Let

By Theorem

Let

Let

By Theorem

Let

We call

Let

By Theorem

Suppose that

Let

By Theorem

Suppose that

Let

By Theorem

Suppose

Let

Suppose that

Suppose

We call

Let

(1)

(2)

(3)

The Pawlak rough sets on the algebraic sets such as semigroups, groups, rings, modules, and lattices were mainly studied by a congruence relation. However, the generalized Pawlak rough set was defined for two universes and proposed on generalized binary relations. Can we extended congruence relations to two universes for algebraic sets? Therefore, Davvaz [

The authors are grateful to the reviewers for their valuable suggestions to improve the paper. This work was supported by the National Science Foundation of China (no. 11071061).