Mater rediens filium salutat.

Because mater is in the nominative singular, the case normally used for the subject of a sentence.

• mater = mother
• nominative singular of a 3rd-declension noun

So mater is the one doing the action.

Rediens is the present active participle of redeo, redire (to return, go back).

A participle is a verbal adjective: it comes from a verb, but it behaves like an adjective. Here it describes mater.

So mater rediens means something like:

• the mother returning
• the returning mother
• the mother, as she returns

It shows an action happening at the same time as the main verb.

Because rediens is a 3rd-declension participle, and in the nominative singular it has the same form for masculine and feminine.

So:

• masculine nominative singular: rediens
• feminine nominative singular: rediens
• neuter nominative singular: rediens

It is still feminine in meaning here because it agrees with mater, which is feminine.

We know by agreement.

A participle agrees with the noun it modifies in:

• gender
• number
• case

Here:

• mater = nominative singular, feminine
• rediens = nominative singular, able to agree with a feminine noun

So rediens modifies mater, not filium.

It also makes sense logically: the mother is the one returning.

Because filium is in the accusative singular, which is the normal case for a direct object.

• filius = son (nominative)
• filium = son as the object (accusative)

So filium is the person being greeted, not the one doing the greeting.

Salutat is the verb saluto, salutare (to greet), and this form is:

• 3rd person singular
• present tense
• active voice
• indicative mood

So it means he/she greets or is greeting.

Since the subject is mater, the sense is the mother greets.

Latin often leaves out possessive words like his, her, or their when the relationship is obvious from context.

So filium can naturally mean:

• the son
• her son

If the Latin wanted to be more explicit, it could say filium suum. But very often that is unnecessary.

Classical Latin has no articles like English the or a/an.

So:

• mater can mean mother, a mother, or the mother
• filium can mean son, a son, or the son

You decide from context which English wording fits best.

The present participle usually shows an action happening at the same time as the main verb.

So the idea is:

• while returning, the mother greets her son
• the mother, returning, greets her son

It does not usually mean that she returned first and then greeted him. For that idea, Latin would normally use a different construction.

Latin word order is more flexible than English because the case endings show each word’s role.

So Mater rediens filium salutat means the same basic thing as:

• Mater filium rediens salutat
• Filium mater rediens salutat
• Mater filium salutat rediens

Though some orders sound more natural or emphasize different words.

In this sentence, the order is quite straightforward:

• mater = subject first
• rediens = describing the subject
• filium = object
• salutat = verb at the end, a very common Latin pattern

Yes. English often has several natural ways to translate a Latin participle.

Possible translations include:

• The returning mother greets her son.
• The mother, returning, greets her son.
• As she returns, the mother greets her son.
• While returning, the mother greets her son.

All of these try to capture the same Latin idea: rediens describes the mother and shows an action happening at the same time as salutat.

You would normally learn them like this:

• mater, matris — mother
• redeo, redire, redii/redivi, reditum — return, go back
• filius, filii — son
• saluto, salutare, salutavi, salutatum — greet

And rediens comes from redeo as its present participle.