# Right self-distributive

## Definition

$$\forall a,b,c \in S : (b \circ c) \circ a = (b \circ a) \circ (c \circ a)$$

**THEOREM**
Let \(\circ\) be idempotent, commutative, and associative. Then \(\circ\) is right self-distributive.

Proof available

**LEMMA**
Union is right self-distributive.

$$(B \cup C) \cup A = (B \cup A) \cup (C \cup A)$$

Proof available

**LEMMA**
Disjunction is right self-distributive.

$$(Q \vee R) \vee P \dashv\vdash (Q \vee P) \vee (R \vee P)$$

Proof available

**LEMMA**
Intersection is right self-distributive.

$$(B \cap C) \cap A = (B \cap A) \cap (C \cap A)$$

Proof available

**LEMMA**
Conjunction is right self-distributive.

$$(Q \wedge R) \wedge P \dashv\vdash (Q \wedge P) \wedge (R \wedge P)$$

Proof available