St Andrews, September 59, 2006 Interpreting graphs in 0-simple - PowerPoint PPT Presentation

St Andrews, September 5–9, 2006 Interpreting graphs in 0-simple semigroups with involution with applications to computational complexity and the finite basis problem Mikhail Volkov (joint work with Marcel Jackson) Ural State University, Ekaterinburg, Russia St Andrews 2006 – p.1/15

Finite Basis Problem: an Overview Now we want to apply our techniques to the Finite Basis Problem (FBP) for unary semigroups. St Andrews 2006 – p.2/15

Finite Basis Problem: an Overview Now we want to apply our techniques to the Finite Basis Problem (FBP) for unary semigroups. First, a few words about “standard” techniques for plain semigroups. St Andrews 2006 – p.2/15

� � � Finite Basis Problem: an Overview Now we want to apply our techniques to the Finite Basis Problem (FBP) for unary semigroups. First, a few words about “standard” techniques for plain semigroups. The techniques are: Syntactic analysis; Critical semigroup method; Use of inherently nonfinitely based semigroups. St Andrews 2006 – p.2/15

✁ ✂ ✁ ✂ ✁ Finite Basis Problem: an Overview Syntactic analysis: given a semigroup , we try first to construct an infinite sequence of identities holding in and then to show that no identity in can be formally deduced from shorter identities holding in . St Andrews 2006 – p.3/15

✄ ✄ ☎ ✄ ☎ Finite Basis Problem: an Overview Syntactic analysis: given a semigroup , we try first to construct an infinite sequence of identities holding in and then to show that no identity in can be formally deduced from shorter identities holding in . Based on Birkhoff’s completeness theorem for equational logic, 1935. St Andrews 2006 – p.3/15

✆ ✆ ✝ ✆ ✝ ✞ Finite Basis Problem: an Overview Syntactic analysis: given a semigroup , we try first to construct an infinite sequence of identities holding in and then to show that no identity in can be formally deduced from shorter identities holding in . Based on Birkhoff’s completeness theorem for equational logic, 1935. First application: by Perkins in ✟✡✠ the 60s to the Brandt monoid . St Andrews 2006 – p.3/15

✓ ✓ ✏ ✓ ✕ ✏ ✓ ✏ ✓ ✏ ✖ ✓ ✏ ✌ ✖ ✓ ✏ ✎ ☛ ✖ ✓ ☞ ✎ ☛ ✖ ☞ ✗ ✘ ☛ ✔ Finite Basis Problem: an Overview Syntactic analysis: given a semigroup , we try first to construct an infinite sequence of identities holding in and then to show that no identity in can be formally deduced from shorter identities holding in . Based on Birkhoff’s completeness theorem for equational logic, 1935. First application: by Perkins in ✍✡✎ the 60s to the Brandt monoid . 3 1 ✏✒✑ ✏✒✑ ✏✒✑ 2 St Andrews 2006 – p.3/15 ✏✒✑

✥ ★ ✦ ✣ ✣ ✥ ✣ ✥ ✥ ✣ ★ ✥ ✣ ✥ ✛ ✣ ✥ ✢ ★ ✙ ✥ ✢ ✚ ★ ✙ ✩ ✚ ✪ ✙ ✙ ✧ Finite Basis Problem: an Overview Syntactic analysis: given a semigroup , we try first to construct an infinite sequence of identities holding in and then to show that no identity in can be formally deduced from shorter identities holding in . Based on Birkhoff’s completeness theorem for equational logic, 1935. First application: by Perkins in ✜✡✢ the 60s to the Brandt monoid . 3 1 ✣✒✤ ✣✒✤ ✣✒✤ 2 A difficulty: ID-CHECK( ) may be hard. St Andrews 2006 – p.3/15 ✣✒✤

✴ ✲ ✫ ✫ ✴ ✬ ✳ ✲ ✬ ✫ ✳ Finite Basis Problem: an Overview Critical semigroup method: given , try to construct for each sufficiently large a semigroup such that ✫✮✭ but every -generated subsemigroup of ✯✱✰ ✫✮✭ ✫✮✭ belongs to . St Andrews 2006 – p.4/15

✼ ✺ ✵ ✵ ✼ ✶ ✻ ✺ ✶ ✵ ✻ Finite Basis Problem: an Overview Critical semigroup method: given , try to construct for each sufficiently large a semigroup such that ✵✮✷ but every -generated subsemigroup of ✸✱✹ ✵✮✷ ✵✮✷ belongs to . Based on Birkhoff’s finite basis theorem, 1935. St Andrews 2006 – p.4/15

✾ ❃ ✽ ❅ ✽ ✾ ❄ ❃ ❂ ✽ ❄ ❂ Finite Basis Problem: an Overview Critical semigroup method: given , try to construct for each sufficiently large a semigroup such that ✽✮✿ but every -generated subsemigroup of ❀✱❁ ✽✮✿ ✽✮✿ belongs to . Based on Birkhoff’s finite basis theorem, 1935. First application: by Kleiman in the 70s to the Brandt ❆❈❇ monoid . St Andrews 2006 – p.4/15

❉ ■ ❑ ❏ ■ ▲ ❊ ❉ ❑ ▲ ❏ ❊ ❉ Finite Basis Problem: an Overview Critical semigroup method: given , try to construct for each sufficiently large a semigroup such that ❉✮❋ but every -generated subsemigroup of ●✱❍ ❉✮❋ ❉✮❋ belongs to . Based on Birkhoff’s finite basis theorem, 1935. First application: by Kleiman in the 70s to the Brandt ▼❈◆ ▼❈◆ monoid . This way Kleiman proved that is nonfinitely based also as an inverse semigroup. St Andrews 2006 – p.4/15

❯ ❲ ❱ ❖ ❱ ❯ ❚ ❖ ❲ ❚ ❚ ❯ P ❱ ❖ ❖ P Finite Basis Problem: an Overview Critical semigroup method: given , try to construct for each sufficiently large a semigroup such that ❖✮◗ but every -generated subsemigroup of ❘✱❙ ❖✮◗ ❖✮◗ belongs to . Based on Birkhoff’s finite basis theorem, 1935. First application: by Kleiman in the 70s to the Brandt ❳❈❨ ❳❈❨ monoid . This way Kleiman proved that is nonfinitely based also as an inverse semigroup. A difficulty: the membership checking for may be hard. St Andrews 2006 – p.4/15

❩ ❭ ❩ ❫ ❫ ❪ ❬ Finite Basis Problem: an Overview Inherently nonfinitely based semigroups: given , try to find an inherently nonfinitely based semigroup within . (A finite semigroup is inherently nonfinitely based if no finitely based locally finite variety can contain .) St Andrews 2006 – p.5/15

❝ ❴ ❵ ❛ ❜ ❴ ❝ Finite Basis Problem: an Overview Inherently nonfinitely based semigroups: given , try to find an inherently nonfinitely based semigroup within . (A finite semigroup is inherently nonfinitely based if no finitely based locally finite variety can contain .) Based on a corollary from Birkhoff’s HSP theorem, 1935. St Andrews 2006 – p.5/15

❤ ❞ ❡ ❢ ❣ ❞ ❤ ✐ Finite Basis Problem: an Overview Inherently nonfinitely based semigroups: given , try to find an inherently nonfinitely based semigroup within . (A finite semigroup is inherently nonfinitely based if no finitely based locally finite variety can contain .) Based on a corollary from Birkhoff’s HSP theorem, 1935. First application: by Sapir in the 80s to the ❥✡❦ Brandt monoid . St Andrews 2006 – p.5/15

♣ ❧ ♠ ♥ ♦ ❧ ♣ q Finite Basis Problem: an Overview Inherently nonfinitely based semigroups: given , try to find an inherently nonfinitely based semigroup within . (A finite semigroup is inherently nonfinitely based if no finitely based locally finite variety can contain .) Based on a corollary from Birkhoff’s HSP theorem, 1935. First application: by Sapir in the 80s to the r✡s Brandt monoid . A difficulty: not so many inherently nonfinitely based semigroups exist (there is a complete classification of them). St Andrews 2006 – p.5/15

② ③ ③ ① t ✉ ③ ② ③ ① ✈ ④ ③ ② ① ✉ ⑥ ① ③ ② Unary Semigroup Case Syntactic analysis: basically fails as combinatorics of unary words becomes rather difficult even in the presence of the law and almost impossible ✉✇✈ ③⑤✈ in the absence of this law – look at . ✉✇✈ ③⑤⑥ St Andrews 2006 – p.6/15

❶ ❷ ❷ ⑦ ❷ ⑩ ⑧ ⑧ ❹ ⑩ ❷ ❶ ⑩ ❸ ❷ ❶ ⑩ ❶ ❷ ⑦ ❺ ⑨ Unary Semigroup Case Syntactic analysis: basically fails as combinatorics of unary words becomes rather difficult even in the presence of the law and almost impossible ⑧✇⑨ ❷⑤⑨ in the absence of this law – look at . ⑧✇⑨ ❷⑤❹ Use of inherently nonfinitely based semigroups: has failed so far as there is no inherently nonfinitely based inverse semigroup (Sapir) and even no inherently nonfinitely based -regular semigroup. St Andrews 2006 – p.6/15

❿ ❿ ❻ ❽ ➂ ❿ ➀ ❾ ❾ ❼ ➀ ➀ ❼ ➀ ❾ ➁ ➀ ❿ ❾ ❻ ➃ ❻ ➄ ➀ Unary Semigroup Case Syntactic analysis: basically fails as combinatorics of unary words becomes rather difficult even in the presence of the law and almost impossible ❼✇❽ ➀⑤❽ in the absence of this law – look at . ❼✇❽ ➀⑤➂ Use of inherently nonfinitely based semigroups: has failed so far as there is no inherently nonfinitely based inverse semigroup (Sapir) and even no inherently nonfinitely based -regular semigroup. Critical semigroup method: works pretty well (Auinger & , still in progress). St Andrews 2006 – p.6/15