'pets-not-allowed' is formed in Modern English from the words 'pet', 'not', and 'allow'. 'Pet' (singular of 'pets') in Modern English means 'a tamed or domesticated animal kept for companionship', 'not' is the negation particle, and 'allow' means 'to permit'.

'pet' developed in Early Modern English (16th century) to mean a favored or domesticated animal; 'not' descends from Old English negation forms (eg. 'nōt'); 'allow' comes via Middle English 'allowen' from Old French (see 'allowen' / 'alouer'), and these words combined in modern usage to form signs such as 'no pets allowed' or the hyphenated notice 'pets-not-allowed'.

The individual words originally had the same basic meanings (pet = favored/domesticated animal; allow = permit), and over time the fixed phrase came to serve as a conventional short notice meaning 'pets are not permitted' with little change in sense.