
mögen
to like
To feel affection or sympathy for someone or something, or to have a positive preference for them.
Ich mag unsere neue Nachbarin sehr.
I like our new neighbor very much.
to like / may
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to like
To feel affection or sympathy for someone or something, or to have a positive preference for them.
Ich mag unsere neue Nachbarin sehr.
I like our new neighbor very much.

would like
To wish for something or want to do something. This form is often used for polite wishes and requests.
Ich möchte bitte einen Kaffee bestellen.
I would like to order a coffee, please.

to want to
To want or be willing to do something, often in negative sentences.
Sie mag heute nicht über den Streit sprechen.
She doesn't want to talk about the argument today.

may
To express that something may be the case or may be a possible explanation.
Der Fehler mag durch ein technisches Problem entstanden sein.
The error may have been caused by a technical problem.

that may be
To acknowledge that a statement may be true or partly true before introducing a contrasting point.
Das mag sein, aber wir müssen die Zahlen trotzdem noch einmal prüfen.
That may be, but we still need to check the figures again.

may
To express a wish in elevated or ceremonial language.
Möge das neue Jahr euch Gesundheit und Frieden bringen.
May the new year bring you health and peace.
The person or thing toward which the positive feeling is directed is used directly in the accusative. The object can be a person, an object, or an activity expressed as a noun.
Meine Tochter mag ihren neuen Lehrer sehr.
My daughter likes her new teacher very much.
The strength of the preference is often indicated with words such as 'sehr' (very much), 'besonders' (especially), 'wirklich' (really), or 'überhaupt nicht' (not at all). This information usually appears before the object or at the end of the sentence.
Ich mag besonders die ruhigen Straßen in diesem Viertel.
I especially like the quiet streets in this neighborhood.
To express a polite wish, 'möchte' (would like) is often used with an accusative object or an infinitive without 'zu' (to). This pattern is especially common when ordering, shopping, and having polite conversations.
Wir möchten für Samstagabend einen Tisch reservieren.
We would like to reserve a table for Saturday evening.
In this usage pattern, the second infinitive appears at the end of the sentence without 'zu' (to). Negation with 'nicht' (not) usually refers to the willingness to perform the action.
Er mag vor den anderen nicht über seine Sorgen reden.
He doesn't want to talk about his worries in front of the others.
This usage is mainly found in factual or elevated texts. When making assumptions about the past, a participle frequently appears together with 'haben' (to have) or 'sein' (to be) at the end of the sentence.
Die Verzögerung mag durch das schlechte Wetter verursacht worden sein.
The delay may have been caused by the bad weather.
In 'Das mag sein' (That may be), 'das' (that) refers to a previous statement. It is often followed by a clause with 'aber' (but) or 'doch' (however) that introduces an objection or qualification.
Das mag sein, doch ohne schriftliche Bestätigung können wir nichts entscheiden.
That may be, but without written confirmation, we cannot make a decision.
In this wish pattern, the form of 'mögen' (may) often comes first, immediately followed by the subject; the infinitive appears at the end of the sentence. This construction is typical of speeches, congratulations, blessings, and ceremonial texts.
Möge diese Entscheidung allen Beteiligten neue Hoffnung geben.
May this decision give everyone involved new hope.
to be able to
Können presents a possible explanation neutrally as a supposition. Mögen often acknowledges a possibility in more formal language and frequently introduces a contrast. Use können for a simple supposition; mögen is suitable for a concession that limits or qualifies what follows.

Die Verzögerung mag am Wetter liegen, doch wir müssen trotzdem weiterfahren.
The delay may be due to the weather, but we still have to keep driving.
Die Verzögerung könnte am Wetter liegen.
The delay could be due to the weather.