Fyrirtækið rekur búðina.

The dictionary forms are:

• fyrirtæki = company, business
• reka = to run, operate, manage
• búð = shop, store

In the sentence, the words appear in inflected forms:

• fyrirtækið = the company
• rekur = runs / operates
• búðina = the shop / the store

So you usually need to strip off endings to find the base form in a dictionary.

Because Icelandic usually adds the definite article onto the end of the noun instead of using a separate word like English the.

• fyrirtæki = a company / company
• fyrirtækið = the company

Here, -ið is the definite ending for this form of the noun.
So fyrirtækið is not a different word; it is the same noun with the attached.

Because búðina is the accusative definite singular form, while búðin is the nominative definite singular form.

That matters because in this sentence búðina is the direct object of the verb rekur.

A quick comparison:

• búðin = the shop as a subject
• búðina = the shop as an object

So:

• Búðin lokar. = The shop closes.
• Fyrirtækið rekur búðina. = The company runs the shop.

English does not show this difference on nouns, but Icelandic does.

You can tell from both word order and case.

1. Word order

The normal order here is:

• Fyrirtækið = subject
• rekur = verb
• búðina = object

So it follows a basic subject–verb–object pattern.

2. Case endings

The noun forms also help:

• fyrirtækið is in the nominative form, which is the usual subject case
• búðina is in the accusative form, which is commonly used for direct objects

So even if word order changes, the endings still give useful clues.

Rekur is the 3rd person singular present tense form of reka.

In this sentence it means:

• runs
• operates
• manages

Because the subject fyrirtækið is singular, the verb is also singular:

• fyrirtækið rekur = the company runs

Because reka is the infinitive form, meaning to run / to operate.

In a normal sentence, Icelandic uses a conjugated verb, not the infinitive:

• reka = to run
• rekur = runs

So:

• Fyrirtækið rekur búðina. = correct
• Fyrirtækið reka búðina. = not correct in standard Icelandic

This is similar to English:

• to run
• the company runs

Reka is a very common verb with several meanings depending on context. It can mean things like:

• run / operate / manage a business
• drive something forward
• chase
• drift in some contexts
• even dismiss/fire someone in certain expressions

In Fyrirtækið rekur búðina, the business meaning is the natural one:
The company operates the shop.

So this is a good example of how Icelandic verbs often have a wider range than a single English translation suggests.

Very often, yes. The normal definite article in Icelandic is usually a suffix attached to the noun.

Examples:

• fyrirtæki → fyrirtækið = the company
• búð → búðin / búðina = the shop in different cases

There is also a separate form like hinn / hin / hið, but that is used in more special, formal, or stylistic situations. For everyday sentences like this one, the suffixed article is the standard pattern.

Yes, it matters because Icelandic noun endings depend heavily on gender, number, and case.

In this sentence:

• fyrirtæki is neuter
• búð is feminine

That helps explain the different endings:

• fyrirtækið: neuter definite singular nominative
• búðina: feminine definite singular accusative

So the endings are not random; they reflect the grammar of each noun.

No, not in this form.

The verb rekur is 3rd person singular, and Icelandic does not use different present-tense verb forms for he/she/it the way the noun system changes for gender.

So whether the singular subject is masculine, feminine, or neuter, you still get the same verb form:

• maðurinn rekur... = the man runs...
• konan rekur... = the woman runs...
• fyrirtækið rekur... = the company runs...

The noun changes by gender; the present-tense verb form here does not.

Yes. Icelandic allows more flexibility than English, because case endings help show who is doing what.

The most neutral order is:

• Fyrirtækið rekur búðina.

But you may also see something like:

• Búðina rekur fyrirtækið.

That still means roughly The company runs the shop, because búðina is still marked as the object and fyrirtækið as the subject.

However, the changed order can sound more marked or emphatic, so the original sentence is the most straightforward version for a learner.