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◆❡✐❣❤❜♦✉r❤♦♦❞ ♣r♦❞✉❝ts ♦❢ ♣r❡tr❛♥s✐t✐✈❡ ❧♦❣✐❝s ✇✐t❤ ❙✺ ❆♥❞r❡② ❑✉❞✐♥♦✈ ■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s✱ ▼♦s❝♦✇ ◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t② ❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s✱ ▼♦s❝♦✇ ▼♦s❝♦✇ ■♥st✐t✉t❡ ♦❢ P❤②s✐❝s ❛♥❞ ❚❡❝❤♥♦❧♦❣② ❙❡♣t❡♠❜❡r ✶✱ ✷✵✶✻

▲❛♥❣✉❛❣❡ ❛♥❞ ❧♦❣✐❝s φ ::= p | ¬ φ | φ ∨ φ | � i φ, i = 1 , 2 . ⊥ ✱ → ❛♥❞ ♦ i ❛r❡ ❡①♣r❡ss✐❜❧❡ ✐♥ t❤❡ ✉s✉❛❧ ✇❛②✳ ◆♦r♠❛❧ ♠♦❞❛❧ ❧♦❣✐❝✳ K n ❞❡♥♦t❡s t❤❡ ♠✐♥✐♠❛❧ ♥♦r♠❛❧ ♠♦❞❛❧ ❧♦❣✐❝ ✇✐t❤ n ♠♦❞❛❧✐t✐❡s ❛♥❞ K = K 1 ✳ L 1 ❛♥❞ L 2 ✖ t✇♦ ♠♦❞❛❧ ❧♦❣✐❝s ✇✐t❤ ♦♥❡ ♠♦❞❛❧✐t② � t❤❡♥ t❤❡ ❢✉s✐♦♥ ♦❢ t❤❡s❡ ❧♦❣✐❝s ✐s ❞❡✜♥❡❞ ❛s L 1 ∗ L 2 = K 2 + L ′ ′ ; 1 + L 2 ✇❤❡r❡ L ′ i ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❢♦r♠✉❧❛s ❢r♦♠ L i ✇❤❡r❡ ✐♥ ❛❧❧ ❢♦r♠✉❧❛s � ✐s r❡♣❧❛❝❡❞ ❜② � i ✳

❚♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ❙❡♠❛♥t✐❝s ♦♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ❝❛♥ ❜❡ ❜✉✐❧t ✉s✐♥❣ ❝❧♦s✉r❡ ♦♣❡r❛t♦r cl ✇❤❡r❡ cl ( A ) ✐s t❤❡ ❝❧♦s✉r❡ ♦❢ A ✳ ❚❤❡ s❡♠❛♥t✐❝s ❞❡✜♥❡❞ ❧✐❦❡ t❤✐s✿ V cl ( ♦ φ ) = cl ( V cl ( φ )) ❖r ✉s✐♥❣ ❞❡r✐✈❛t✐✈❡ ♦♣❡r❛t♦r d ✱ ✇❤❡r❡ d ( A ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❧✐♠✐t ♣♦✐♥ts ♦❢ A ✳ ❚❤❡ s❡♠❛♥t✐❝s ❞❡✜♥❡❞ ❧✐❦❡ t❤✐s✿ V d ( ♦ φ ) = d ( V d ( φ )) ❝❧♦s✉r❡ s❡♠❛♥t✐❝s ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ❛❧❧ s♣❛❝❡s S4 ✭▼❝❑✐♥s❡② ✫ ❚❛rs❦✐✬✶✾✹✹✮ wK4 ✭❊s❛❦✐❛✬✶✾✽✶✮ Q ✱ ❈❛♥t♦r s♣❛❝❡ S4 D4 ✭❙❤❡❤t♠❛♥✬✶✾✾✵✮ R S4 D4 + G 2 ✭❙❤❡❤t♠❛♥✬✷✵✵✵✮ R n , n ≥ 2 S4 D4 + G 1 ✭❙❤❡❤t♠❛♥✬✶✾✾✵✮ wK4 = K + ♦♦ p → ♦ p ∨ p D4 = K + ♦♦ p → ♦ p + ♦⊤

❚❤❡ ♣r♦❞✉❝t ♦❢ ❑r✐♣❦❡ ❢r❛♠❡s ❋♦r t✇♦ ❢r❛♠❡s F 1 = ( W 1 , R 1 ) ❛♥❞ F 2 = ( W 2 , R 2 ) F 1 × F 2 = ( W 1 × W 2 , R ∗ 1 , R ∗ 2 ) , ✇❤❡r❡ ( a 1 , a 2 ) R ∗ 1 ( b 1 , b 2 ) ⇔ a 1 R 1 b 1 & a 2 = b 2 ( a 1 , a 2 ) R ∗ 2 ( b 1 , b 2 ) ⇔ a 1 = b 1 & a 2 R 2 b 2 ❋♦r t✇♦ ❧♦❣✐❝s L 1 ❛♥❞ L 2 L 1 × L 2 = Log ( { F 1 × F 2 | F 1 | = L 1 & F 2 | = L 2 } ) ✭❙❤❡❤t♠❛♥✱ ✶✾✼✽✮ ❋♦r t✇♦ ❝❧❛ss❡s ♦❢ ❢r❛♠❡s F 1 ❛♥❞ F 2 Log ( { F 1 × F 2 | F 1 ∈ F 1 & F 2 ∈ F 2 } ) ⊇ Log ( F 1 ) ∗ Log ( F 2 )+ + � 1 � 2 p ↔ � 1 � 2 p + ♦ 1 � 2 p → � 2 ♦ 1 p ✳ K × K = K ∗ K + � 1 � 2 p ↔ � 1 � 2 p + ♦ 1 � 2 p → � 2 ♦ 1 p S4 × S4 = S4 ∗ S4 + � 1 � 2 p ↔ � 1 � 2 p + ♦ 1 � 2 p → � 2 ♦ 1 p ✳ ✳ ✳

❚❤❡ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ ❋♦r t✇♦ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X 1 = ( X 1 , τ 1 ) ❛♥❞ X 2 = ( X 2 , τ 2 ) X 1 × X 2 = ( X 1 × X 2 , τ ∗ 1 , τ ∗ 2 ) , ✇❤❡r❡ τ ∗ 1 ❤❛s ❜❛s❡ { U 1 × x 2 | U 1 ∈ τ 1 & x 2 ∈ X 2 } τ ∗ 2 ❤❛s ❜❛s❡ { x 1 × U 2 | x 1 ∈ X 1 & U 2 ∈ τ 2 }

❚❤❡ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ ❋♦r t✇♦ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X 1 = ( X 1 , τ 1 ) ❛♥❞ X 2 = ( X 2 , τ 2 ) X 1 × X 2 = ( X 1 × X 2 , τ ∗ 1 , τ ∗ 2 ) , ✇❤❡r❡ τ ∗ 1 ❤❛s ❜❛s❡ { U 1 × x 2 | U 1 ∈ τ 1 & x 2 ∈ X 2 } τ ∗ 2 ❤❛s ❜❛s❡ { x 1 × U 2 | x 1 ∈ X 1 & U 2 ∈ τ 2 } y x

❚❤❡ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ ❋♦r t✇♦ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X 1 = ( X 1 , τ 1 ) ❛♥❞ X 2 = ( X 2 , τ 2 ) X 1 × X 2 = ( X 1 × X 2 , τ ∗ 1 , τ ∗ 2 ) , ✇❤❡r❡ τ ∗ 1 ❤❛s ❜❛s❡ { U 1 × x 2 | U 1 ∈ τ 1 & x 2 ∈ X 2 } τ ∗ 2 ❤❛s ❜❛s❡ { x 1 × U 2 | x 1 ∈ X 1 & U 2 ∈ τ 2 } ❋♦r t✇♦ ❧♦❣✐❝s L 1 ❛♥❞ L 2 L 1 × t L 2 = Log ( { X 1 × X 2 | X 1 | = L 1 & X 2 | = L 2 } S4 × t S4 = Log ( Q × Q ) = S4 ∗ S4 ✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ Log ( R × R ) � = S4 ∗ S4 ✭❑r❡♠❡r✱ ✷✵✶✵❄✮ Log ( Cantor × Cantor ) � = S4 ∗ S4 ❞✲❧♦❣✐❝ ♦❢ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇❛s ❝♦♥s✐❞❡r❡❞ ❜② ▲✳ ❯r✐❞✐❛ ✭✷✵✶✶✮✳ Log d ( Q × Q ) = D4 ∗ D4 ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♥❡✐❣❤❜♦r❤♦♦❞ ❢r❛♠❡s ✇❛s ❞♦♥❡ ❜② ❑✳ ❙❛♥♦ ✭✷✵✶✶✮✳

❑♥♦✇♥ r❡s✉❧ts ❚❤❡♦r❡♠ ✭✷✵✶✷✮ ▲❡t L 1 ❛♥❞ L 2 ❜❡ ❢r♦♠ t❤❡ s❡t { D , T , D4 , S4 } t❤❡♥ L 1 × n L 2 = L 1 ∗ L 2 . ◆♦t str❛✐❣❤t❢♦r✇❛r❞ ❜✉t st✐❧❧ ❛ ❈♦r♦❧❧❛r② ■♥ ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✶✳ D4 × d D4 = D4 ∗ D4 ✳ ✷✳ ❬❯r✐❞✐❛✬✷✵✶✶❪ Log d ( Q × Q ) = D4 ∗ D4 ◆♦t❡ t❤❛t ❛❧❧ t❤❡s❡ ❧♦❣✐❝s ✐♥❝❧✉❞❡ s❡r✐❛❧✐t②✿ ¬ � ⊥ ✳

❑♥♦✇♥ r❡s✉❧ts ❚❤❡♦r❡♠ ✭✷✵✶✷✮ ▲❡t L 1 ❛♥❞ L 2 ❜❡ ❢r♦♠ t❤❡ s❡t { D , T , D4 , S4 } t❤❡♥ L 1 × n L 2 = L 1 ∗ L 2 . ◆♦t str❛✐❣❤t❢♦r✇❛r❞ ❜✉t st✐❧❧ ❛ ❈♦r♦❧❧❛r② ■♥ ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✶✳ D4 × d D4 = D4 ∗ D4 ✳ ✷✳ ❬❯r✐❞✐❛✬✷✵✶✶❪ Log d ( Q × Q ) = D4 ∗ D4 ◆♦t❡ t❤❛t ❛❧❧ t❤❡s❡ ❧♦❣✐❝s ✐♥❝❧✉❞❡ s❡r✐❛❧✐t②✿ ¬ � ⊥ ✳

❲✐t❤♦✉t s❡r✐❛❧✐t② ■t ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r ❧♦❣✐❝ K ✦ ▲❡♠♠❛ ❋♦r ❛♥② t✇♦ ♥✲❢r❛♠❡s X 1 ❛♥❞ X 2 X 1 × X 2 | = � 1 ⊥ → � 2 � 1 ⊥ . ❆♥❞ ❡✈❡♥ ♠♦r❡✱ ❢♦r ❛♥② ❝❧♦s❡❞ � 1 ✲❢r❡❡ ❢♦r♠✉❧❛ φ ❛♥❞ ❛♥② ❝❧♦s❡❞ � 2 ✲❢r❡❡ ❢♦r♠✉❧❛ ψ X 1 × X 2 | = φ → � 1 φ, X 1 × X 2 | = ψ → � 2 ψ. Pr♦♦❢✳ ⇒ ∅ ∈ τ ′ X 1 × X 2 , ( x, y ) | = � 1 ⊥ ⇐ 1 ( x, y ) ⇐ ⇒ ⇒ ∀ y ′ ∈ X 2 ( ∅ ∈ τ ′ 1 ( x, y ′ )) ⇐ ∅ ∈ τ 1 ( x ) ⇐ ⇒ ∀ y ′ ∈ X 2 ( X 1 × X 2 , ( x, y ′ ) | = � 1 ⊥ ) = ⇒ X 1 × X 2 , ( x, y ) | = � 2 � 1 ⊥ . ❍❡♥❝❡✱ X 1 × X 2 | = � 1 ⊥ → � 2 � 1 ⊥ ✳

❲✐t❤♦✉t s❡r✐❛❧✐t② ■t ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r ❧♦❣✐❝ K ✦ ▲❡♠♠❛ ❋♦r ❛♥② t✇♦ ♥✲❢r❛♠❡s X 1 ❛♥❞ X 2 X 1 × X 2 | = � 1 ⊥ → � 2 � 1 ⊥ . ❆♥❞ ❡✈❡♥ ♠♦r❡✱ ❢♦r ❛♥② ❝❧♦s❡❞ � 1 ✲❢r❡❡ ❢♦r♠✉❧❛ φ ❛♥❞ ❛♥② ❝❧♦s❡❞ � 2 ✲❢r❡❡ ❢♦r♠✉❧❛ ψ X 1 × X 2 | = φ → � 1 φ, X 1 × X 2 | = ψ → � 2 ψ. Pr♦♦❢✳ ❙✐♥❝❡ ψ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ♥❡✐t❤❡r � 2 ✱ ♥♦r ✈❛r✐❛❜❧❡s✱ ✐ts ✈❛❧✉❡ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ s❡❝♦♥❞ ❝♦♦r❞✐♥❛t❡✳ ▲❡t F = X 1 × X 2 ✳ ❙♦ F, ( x, y ) | = ψ ✱ t❤❡♥ ∀ y ′ ( F, ( x, y ′ ) | = ψ ) ✱ ❤❡♥❝❡✱ F, ( x, y ) | = � 2 ψ ✳

❲✐t❤♦✉t s❡r✐❛❧✐t② ▲❡♠♠❛ ❋♦r ❛♥② t✇♦ ♥✲❢r❛♠❡s X 1 ❛♥❞ X 2 X 1 × X 2 | = � 1 ⊥ → � 2 � 1 ⊥ . ❆♥❞ ❡✈❡♥ ♠♦r❡✱ ❢♦r ❛♥② ❝❧♦s❡❞ � 1 ✲❢r❡❡ ❢♦r♠✉❧❛ φ ❛♥❞ ❛♥② ❝❧♦s❡❞ � 2 ✲❢r❡❡ ❢♦r♠✉❧❛ ψ X 1 × X 2 | = φ → � 1 φ, X 1 × X 2 | = ψ → � 2 ψ. ❉❡✜♥✐t✐♦♥ ❋♦r t✇♦ ✉♥✐♠♦❞❛❧ ❧♦❣✐❝s L 1 ❛♥❞ L 2 ✱ ✇❡ ❞❡✜♥❡ ✇❡❛❦ ❝♦♠♠✉t❛t♦r � L 1 , L 2 � = L 1 ∗ L 2 + ∆ , ✇❤❡r❡ ∆ = { φ → � 2 φ | φ ✐s ❝❧♦s❡❞ ❛♥❞ � 2 ✲❢r❡❡ }∪{ ψ → � 1 ψ | ψ ✐s ❝❧♦s❡❞ ❛♥❞ � 1 ✲❢r❡❡ } . ▲❡♠♠❛ ❋♦r ❛♥② t✇♦ ♥♦r♠❛❧ ♠♦❞❛❧ ❧♦❣✐❝s L 1 ❛♥❞ L 2 � L 1 , L 2 � ⊆ L 1 × n L 2 ✳ ◆♦t❡ t❤❛t ✐❢ ♦⊤ ∈ L 1 ∩ L 2 t❤❡♥ L 1 ∗ L 2 | = ∆ ✳