1. Introduction

The definition of homotopy theory for topological semigroups and most of the backgrounds for this paper have been worked out previously by Čerin in [1]. He introduced the concepts of -homotopy relation, pathwise -connectedness, -homotopy domination, -contractibility, and -fibration.

A topological semigroup is called an -subspace of a topological semigroup if is a subspace of , the map takes the product into and for all . The pair is a topological semigroup with the compact-open topology on a path space , where the multiplication on defined by

will denote a topological semigroup . For , will denote the constant path into in . The function is called an -map of into if is a continuous map of space into such that

Two -maps are called -homotopic (written ) if there is an -map (called -homotopy) such that and for all , for more details see [1–3].

Recall [1] that a surjective -map is called -fibration if it has an -homotopy lifting property for any element in , where is the collection of all topological semigroup. Now we introduce the concept of an -lifting function of an -fibration as follows: let be an -map. Then the function is called an -lifting function for if it is satisfying the following:(1) is an -map,(2) for ,(3) for ,

where . If for all , then the -lifting function is called an -regular lifting function. Also we say that the -map is an -regular fibration if it has -regular lifting function.

Note that we can easily prove that -map is an -regular fibration if and only if it has -regular lifting function. Further, if we let be an -fibration and be an -subspace of . Then -map is an -fibration, where will be the -fibration . Thus, one by defining two -fibrations and , an -map and setting the relation , then we can also study two -fiber maps and whether they are -fiber homotopic.

Theorem 1.1 (See [4]). Let , , and be topological spaces. Then the map always gives rise to a map by defining for all . If is locally compact and regular space, then the map always gives rise to map by defining for all .

By the natural topological semigroup, we mean a topological semigroup , where is continuous associative multiplication on defined by and for all .

2. -Homotopy Extension Property

In this section, we extend the notions of an absolute retract (AR), an absolute neighborhood retract (ANR) and homotopy extension property in homotopy theory for topological spaces into their analogical structure in homotopy theory for topological semigroups. Also we show the relations among them.

Definition 2.1 (See [1]). Let be an -subspace of topological semigroup . The -retraction of onto is an -map such that for all . When there is an -retraction of onto , then we say that is an -retract of .

Definition 2.2. A topological semigroup is called an -absolute retract (-AR) for normal topological semigroups in the class if for every closed -subspace of , any -map has an extension -map .

Definition 2.3. A topological semigroup is called an -absolute neighborhood retract (-ANR) for normal topological semigroups in the class if for every closed -subspace of , any -map can be extended to an open neighborhood -subspace of .

By using the above Definitions 2.2 and 2.3, we can easily state the following theorem.

Theorem 2.4. If is an -AR (resp. -ANR), then is an AR (resp. ANR).

The converse of Theorem 2.4 above needs not to be true for example: If denotes the usual multiplication on the the unit closed interval . denotes the first half of . It is clear that the topological semigroup is a closed -subspace of a normal topological semigroup . Also it is clear that is an ANR. Take and in Definition 2.3, let the identity -map has an extension -map , where is an open neighborhood -subspace of in . In the relative usual topology on , we can get an element and , that is, . Since and an -map would have to satisfy an impossible condition:

that implies . Hence is not -ANR. Similarly, is an AR that but it is not -AR.

The converses of Theorem 2.4 are true for the collection of natural topological semigroups as shown in the following theorem.

Theorem 2.5. A topological semigroup is an -AR (resp. an -ANR) for normal topological semigroups if and only if the topological space is AR (resp. ANR).

Proof. It is clear that the function is an -map if and only if the function is a continuous. Hence a topological semigroup is an -AR (resp. an -ANR) for normal topological semigroups if and only if the topological space is AR (resp. ANR).

Theorem 2.6. Ifis an-AR, thenis also an-AR.

Proof. Let be a closed -subspace of a normal topological semigroup and let be an -map. Since is a locally compact and regular, then by Theorem 1.1, for , we can define an -map by Since is an -AR, then, for , the -map has an extension -map . Hence has an extension -map defined by Hence, is an -AR.

Now in the following definitions, we give the concept of an -homotopy extension property in homotopy theory for topological semigroups.

Definition 2.7. Let be a topological semigroup and be an -subspace of . We will mean by -maps for     in   with respect to a topological semigroup the two -maps such that

Definition 2.8. Let and be two topological semigroups. A closed -subspace of is said to have -homotopy extension property in   with respect to if any -maps can be extended to an -homotopy . That is,

Theorem 2.9. If a closed -subspace of has -homotopy extension property in with respect to , then has homotopy extension property in a space with respect to a space .

Proof. Let be any continuous map. Define -maps for in with respect to a topological semigroup by and for all , , . Then by the hypothesis -maps can be extended to an -homotopy . That is, Since is a locally compact and regular space, then by Theorem 1.1 the function which is defined by for all is continuous. Hence is an extension of . That is, has homotopy extension property in a space with respect to a space .

In the following theorem, we clarify that any closed -subspace of a normal topological semigroup has -homotopy extension property in with respect to any -AR space .

Theorem 2.10. Let be an -AR and be a closed -subspace of a normal topological semigroup . Then has -homotopy extension property in with respect to .

Proof. Let that there is -maps. Since is a locally compact and regular, then by Theorem 1.1, for , define an -map by Since is an -AR, then for , the -map has an extension -map . While at the case , we can take . Hence there is an -map defined by Also we observe that That is, has -homotopy extension property in with respect to .

3. -Homotopy Extension Theorems

The general extension theorem for maps of any closed subsets of space into space discusses under which conditions on and , for every closed , each map will be extendable over relative to . The concepts of an AR, an ANR, and homotopy extension property support this theorem in homotopy theory for topological spaces. Now in this section, we introduce the notions of homotopy extension theorem and fiber homotopy extension theorem in homotopy theory for topological semigroups via -AR property.

Lemma 3.1. Let be a topological semigroup. For each path in and , let be a path in defined by for all , where be any map. Then the function defined by is an -map.

Proof. By Theorem 1.1 to prove that is continuous, it is sufficient to prove that the function defined by for all , is continuous. Let and be a neighborhood of in . Then . By the continuity of and is an open set containing , there is open set in such that and . Also by the continuity of , there are two open sets and in such that Since is an open set in containing , then there is a positive number such that Since and are compact sets in and is a continuous, then is also a compact set in . Now consider that a neighborhood of in . Hence for That is, . Hence the function is continuous.
Now for , Hence , that is, is an -map.

The following theorem clarifies the notion of -fiber homotopy extension theorem in homotopy theory for topological semigroups via -AR property

Theorem 3.2. Let be an -regular fibration with an -AR . Let be a closed -subspace of a normal topological semigroup . If there is -maps such that then there is an -map such that is an extension of , and

Proof. Since is an -AR and is a closed -subspace of a normal topological semigroup , then, by Theorem 2.10, the -map can be extended to -map such that . Now for and , we can define the path in by for all . Then by Lemma 3.1, the function is an -map. Hence, we can define the -map by Firstly, we show that . By the -regularity of and since is an extension for , we observe that for , Secondly, we show that is an extension of . Since is an extension for and by the hypothesis, we get that for all and . Hence, by the -regularity of , we get that Hence is an extension for . Finally, we also observe that for all , .

Now we can give the other rephrasing of above theorem in the following corollary.

Corollary 3.3. Let be an -regular fibration with an -AR and be a closed -subspace of a normal topological semigroup . Let be two -maps and be an -homotopy between them such that If has an extension -map to all of , then has an extension -map to all of . Also there is an -homotopy between and such that is an extension of and

Proof. Since has an extension -map to all of , that is, Hence there is -maps with the property Then by Theorem 3.2, there is an -map such that is an extension of , and Also we can define the -map by for all . Then is an -homotopy between and and also That is, is an extension of .

The following theorem clarifies the notion of -fiber homotopy extension theorem via -AR property.

Theorem 3.4. Let and be two -fibrations with an -AR , a normal and be a closed -subspace of . Let be two -fiber maps between two -fibrations and such that . If is extendable to a full -fiber map , then is extendable to a full -fiber map and .

Proof. Since , then there is an homotopy such that , and Since is extendable to a full -fiber map , then there is -maps. Since is a closed -subspace of a normal , we can apply Theorem 3.2 on -fibration , where and . Hence there is an -map such that is an extension of , and Now we can define by . Hence is an -homotopy between and . From (3.22) and since is an -fiber map, we get that is an -fiber map and Hence.