Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

Weak limits of powers of Chacon’s automorphism

We completely describe the weak closure of the powers of the Koopman operator associated with Chacon’s classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of$\alpha $-weak mixing for Chacon’s automorphism.