
die Aussetzung
abandonment
Leaving behind a person in need of help or an animal in a place where they cannot help themselves.
Die Aussetzung von Haustieren im Wald ist eine Straftat.
The abandonment of pets in the forest is a criminal offense.
abandonment / suspension / reward
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abandonment
Leaving behind a person in need of help or an animal in a place where they cannot help themselves.
Die Aussetzung von Haustieren im Wald ist eine Straftat.
The abandonment of pets in the forest is a criminal offense.

suspension
A temporary interruption or postponement of a proceeding, punishment, or payment.
Das Gericht beschloss die Aussetzung des Verfahrens bis zur Klärung neuer Beweise.
The court decided on the suspension of the proceedings until new evidence had been clarified.

reward / bounty
A reward promised for the capture of a criminal or for a special achievement.
Für Hinweise, die zur Ergreifung des Täters führen, wurde eine hohe Aussetzung versprochen.
A large reward was promised for information leading to the capture of the offender.
'Aussetzung' is a noun derived from the separable verb 'aussetzen' (to abandon; to suspend). The ending '-ung' turns the verb into a feminine noun and describes the action or result of 'Aussetzen' (abandoning or suspending).
Die Aussetzung des Hundes war herzlos.
The abandonment of the dog was heartless.
The three meanings are very different. The context reveals what is meant: if it is about animals or helpless people, it means being left behind. In a legal or formal text, it usually means 'Unterbrechung' (interruption/suspension). If it is about money for information, it means a reward.
Die Firma beantragte eine Aussetzung der Zahlungen.
The company applied for a suspension of payments.
In the meaning 'Zurücklassen' (leaving behind/abandonment), the word is often used in legal or formal contexts. It is a term from criminal law and animal protection law.
Die Aussetzung eines Kindes ist eine schwere Straftat.
The abandonment of a child is a serious criminal offense.
In the meaning 'Unterbrechung' (interruption/suspension), 'Aussetzung' is often used with a genitive that indicates what is being interrupted. Typical phrases are 'Aussetzung des Verfahrens' (suspension of the proceedings), 'Aussetzung der Strafe' (suspension of the sentence), or 'Aussetzung der Vollstreckung' (suspension of enforcement).
Der Anwalt beantragte die Aussetzung des Verfahrens aus gesundheitlichen Gründen.
The lawyer applied for the suspension of the proceedings for health reasons.
postponement / deferment
Aussetzung and Aufschub can both refer to a temporary delay. Aussetzung emphasizes that an ongoing process or obligation is temporarily not carried out. Aufschub, by contrast, emphasizes that an action or appointment is moved to a later point in time.

Die Bank erlaubte eine Aussetzung der Ratenzahlung für drei Monate.
The bank allowed the installment payments to be suspended for three months.
Die Bank gewährte einen Aufschub der nächsten Rate bis September.
The bank granted a deferment of the next installment until September.