Left self-distributive

Definition

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

Let $$\circ$$ be idempotent, commutative, and associative. Then $$\circ$$ is left self-distributive.

Proof available

LEMMA

Conjunction is left self-distributive.

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

Proof available

LEMMA

Disjunction is left self-distributive.

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

Proof available

LEMMA

Union is left self-distributive.

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

Proof available

LEMMA

Join is left self-distributive.

$$x \sqcup (y \sqcup z) = (x \sqcup y) \sqcup (x \sqcup z)$$

Proof available

LEMMA

Intersection is left self-distributive.

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

Proof available

THEOREM

The left operation is left self-distributive.

$$x \leftarrow (y \leftarrow z) = (x \leftarrow y) \leftarrow (x \leftarrow z)$$

Proof available