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By Claudia Casadio, Bob Coecke, Michael Moortgat, Philip Scott (eds.)

For greater than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, type thought, linguistics, theoretical physics, common sense and facts conception. This Festschrift was once prepare at the social gathering of his ninetieth birthday. The papers in it provide a superb photograph of the a number of learn components the place the impression of Jim Lambek's paintings will be felt. the amount contains contributions through well known researchers and through their scholars, displaying how Jim Lambek's principles maintain inspiring upcoming generations of scholars.

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Extra resources for Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday

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CB B 42 A. Bastenhof Definition 10. We define the maps ∥ ⋅ ∥○ and ∥ ⋅ ∥● by mutual induction: ∥p∥○ ∥1∥○ ∥0∥○ ∥A B∥○ ∥A/B∥○ ∥B/A∥○ ∥A B∥○ ∥A B∥○ ∥B A∥○ ∶= ∶= ∶= ∶= ∶= ∶= ∶= ∶= ∶= p ∥p∥● 1 ∥1∥● 0 ∥0∥● ○ ○ ∥A∥ ∥B∥ ∥A B∥● ○ ● ∥A∥ ∥B∥ ∥A/B∥● ○ ∥A/B∥ ∥B/A∥● ○ ○ ∥A∥ ((1/0) ∥B∥ ) ∥A B∥● ○ ∥A∥ (0 ∥B∥● ) ∥A B∥● ○ ∥A B∥ ∥B A∥● ∶= ∶= ∶= ∶= ∶= ∶= ∶= ∶= ∶= 1/p 1 1/0 ∥A∥● ∥B∥● ∥A∥● ∥B∥○ ∥A/B∥● ∥A∥● (∥B∥● 0) ∥A∥● ((1/0) ∥B∥○ ) ∥A B∥● Compare the above definition to the Abelian group interpretation of M&P: multiplications A ⋅ B and inverses A−1 are rendered as A B and 1/A, while 0 replaces the special atom .

Define, by mutual induction, positive and negative contexts: X + [], Y + [] ∶∶= [] ∣ (X + [] B) ∣ (A Y + []) ∣ (X + [] B) ∣ (A Y + []) ∣ (X + []/B) ∣ (B/X + []) ∣ (X + [] B) ∣ (B X + []) ∣ (A/Y − []) ∣ (Y − []/A) ∣ (A Y − []) ∣ (Y − [] A) X − [], Y − [] ∶∶= (X − [] B) ∣ (A Y − []) ∣ (X − [] B) ∣ (A Y − []) ∣ (X − []/B) ∣ (B/X − []) ∣ (X − [] B) ∣ (B X − []) ∣ (A/Y + []) ∣ (Y + []/A) ∣ (A Y + []) ∣ (Y + [] A) 31 32 A. Bastenhof Evidently, given some X + [], Y − [] and f ∶ A → B, we have X + [f ] ∶ X + [A] → X + [B] and Y − [f ] ∶ Y − [B] → Y − [A].

We next survey several definable arrows, proving useful in what is to follow. Definition 4. Lifting is defined / / / lA,B ∶= ((eB,A /iA/B ) ○ hA,A/B ) / lA,B lA,B lA,B : A → B/(A/B) / / ((iB/A /eB,A ) ○ hA,B/A ) ∶= ∶= (hA,A ∶= (hA,B ○ (eB,A A ○ (iB A B : A → (B/A)/A iA B )) : B (A B) → A eB,A )) : (B A) B → A Using the notation introduced in our previous discussion: A h/ (A/B))/(A/B) e/ B/(A/B) (A (B (B A) A) ((B A A (B/A)/((B/A) (B/A)/B B A) e A) h (A B (A (A B)) A A) h/ e/ B) (A B) e h Definition 5.

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