A lower bound for $ x $ and $ y $ is an element $ a \in A $ such that $ a \leq x $ and $ a \leq y $ . A greatest lower bound or meet of $ x $ and $ y $ is a lower bound $ z $ for $ x $ and $ y $ with the further property that whenever $ a $ is a lower bound for $ x $ and $ y $ , we have $ a \leq z $ . When a poset is regarded as a category, meets are exactly products.