Characterizations and classifications of quasitrivial semigroups Jimmy Devillet University of Luxembourg Luxembourg in collaboraton with Jean-Luc Marichal and Bruno Teheux Part I: Single-plateauedness and 2-quasilinearity Weak orderings Recall that a weak ordering (or total preordering) on a set X is a binary relation - on X that is total and transitive. Defining a weak ordering on X amounts to defining an ordered partition of X For X = fa1; a2; a3g, we have 13 weak orderings a1 ≺ a2 ≺ a3 a1 ∼ a2 ≺ a3 a1 ∼ a2 ∼ a3 a1 ≺ a3 ≺ a2 a1 ≺ a2 ∼ a3 a2 ≺ a1 ≺ a3 a2 ≺ a1 ∼ a3 a2 ≺ a3 ≺ a1 a3 ≺ a1 ∼ a2 a3 ≺ a1 ≺ a2 a1 ∼ a3 ≺ a2 a3 ≺ a2 ≺ a1 a2 ∼ a3 ≺ a1 Weak orderings Recall that a weak ordering (or total preordering) on a set X is a binary relation - on X that is total and transitive. Defining a weak ordering on X amounts to defining an ordered partition of X For X = fa1; a2; a3g, we have 13 weak orderings a1 ≺ a2 ≺ a3 a1 ∼ a2 ≺ a3 a1 ∼ a2 ∼ a3 a1 ≺ a3 ≺ a2 a1 ≺ a2 ∼ a3 a2 ≺ a1 ≺ a3 a2 ≺ a1 ∼ a3 a2 ≺ a3 ≺ a1 a3 ≺ a1 ∼ a2 a3 ≺ a1 ≺ a2 a1 ∼ a3 ≺ a2 a3 ≺ a2 ≺ a1 a2 ∼ a3 ≺ a1 Weak orderings Recall that a weak ordering (or total preordering) on a set X is a binary relation - on X that is total and transitive. Defining a weak ordering on X amounts to defining an ordered partition of X For X = fa1; a2; a3g, we have 13 weak orderings a1 ≺ a2 ≺ a3 a1 ∼ a2 ≺ a3 a1 ∼ a2 ∼ a3 a1 ≺ a3 ≺ a2 a1 ≺ a2 ∼ a3 a2 ≺ a1 ≺ a3 a2 ≺ a1 ∼ a3 a2 ≺ a3 ≺ a1 a3 ≺ a1 ∼ a2 a3 ≺ a1 ≺ a2 a1 ∼ a3 ≺ a2 a3 ≺ a2 ≺ a1 a2 ∼ a3 ≺ a1 Single-plateaued weak orderings Definition. (Black, 1948) Let ≤ be a total ordering on X and let - be a weak ordering on X . Then - is said to be single-plateaued for ≤ if ai < aj < ak =) aj ≺ ai or aj ≺ ak or ai ∼ aj ∼ ak Examples. On X = fa1 < a2 < a3 < a4 < a5 < a6g a3 ∼ a4 ≺ a2 ≺ a1 ∼ a5 ≺ a6 a3 ∼ a4 ≺ a2 ∼ a1 ≺ a5 ≺ a6 6 6 a3 ∼ a4 a3 ∼ a4 A A a2 r r a1 ∼ a2 r r A A a1 ∼ a5 r A a5 r r A @ @ a6 r r @ a6 r @ r - r - a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 Single-plateaued weak orderings Definition. (Black, 1948) Let ≤ be a total ordering on X and let - be a weak ordering on X . Then - is said to be single-plateaued for ≤ if ai < aj < ak =) aj ≺ ai or aj ≺ ak or ai ∼ aj ∼ ak Examples. On X = fa1 < a2 < a3 < a4 < a5 < a6g a3 ∼ a4 ≺ a2 ≺ a1 ∼ a5 ≺ a6 a3 ∼ a4 ≺ a2 ∼ a1 ≺ a5 ≺ a6 6 6 a3 ∼ a4 a3 ∼ a4 A A a2 r r a1 ∼ a2 r r A A a1 ∼ a5 r A a5 r r A @ @ a6 r r @ a6 r @ r - r - a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 Single-plateaued weak orderings Q: Given - is it possible to find ≤ for which - is single-plateaued? 0 Example: On X = fa1; a2; a3; a4g consider - and - defined by 0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4 Yes! Consider ≤ defined by a3 < a1 < a2 < a4 6 a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings Q: Given - is it possible to find ≤ for which - is single-plateaued? 0 Example: On X = fa1; a2; a3; a4g consider - and - defined by 0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4 Yes! Consider ≤ defined by a3 < a1 < a2 < a4 6 a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings Q: Given - is it possible to find ≤ for which - is single-plateaued? 0 Example: On X = fa1; a2; a3; a4g consider - and - defined by 0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4 Yes! Consider ≤ defined by a3 < a1 < a2 < a4 6 a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings Q: Given - is it possible to find ≤ for which - is single-plateaued? 0 Example: On X = fa1; a2; a3; a4g consider - and - defined by 0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4 Yes! Consider ≤ defined by a3 < a1 < a2 < a4 6 a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! Single-plateaued weak orderings Q: Given - is it possible to find ≤ for which - is single-plateaued? 0 Example: On X = fa1; a2; a3; a4g consider - and - defined by 0 0 0 a1∼a2≺a3∼a4 and a1≺ a2∼ a3∼ a4 Yes! Consider ≤ defined by a3 < a1 < a2 < a4 6 a1 ∼ a2 J r rJ a3 ∼ a4 J r r - a3 a1 a2 a4 No! 2-quasilinear weak orderings Definition. We say that - is 2-quasilinear if a ≺ b ∼ c ∼ d =) a; b; c; d are not pairwise distinct Proposition Assume the axiom of choice. - is 2-quasilinear () 9 ≤ for which - is single-plateaued 2-quasilinear weak orderings Definition. We say that - is 2-quasilinear if a ≺ b ∼ c ∼ d =) a; b; c; d are not pairwise distinct Proposition Assume the axiom of choice. - is 2-quasilinear () 9 ≤ for which - is single-plateaued Part II: Quasitrivial semigroups Quasitriviality Definition F : X 2 ! X is said to be quasitrivial (or conservative) if F (x; y) 2 fx; yg x; y 2 X Example. F = max≤ on X = f1; 2; 3g endowed with the usual ≤ 6 3 3 r r r 2 2 r r r 1 1 r r r - 1 2 3 Quasitriviality Definition F : X 2 ! X is said to be quasitrivial (or conservative) if F (x; y) 2 fx; yg x; y 2 X Example. F = max≤ on X = f1; 2; 3g endowed with the usual ≤ 6 3 3 r r r 2 2 r r r 1 1 r r r - 1 2 3 Quasitriviality Definition F : X 2 ! X is said to be quasitrivial (or conservative) if F (x; y) 2 fx; yg x; y 2 X Example. F = max≤ on X = f1; 2; 3g endowed with the usual ≤ 6 3 3 r r r 2 2 r r r 1 1 r r r - 1 2 3 Projections Definition. 2 2 The projection operations π1 : X ! X and π2 : X ! X are respectively defined by π1(x; y) = x; x; y 2 X π2(x; y) = y; x; y 2 X Quasitrivial semigroups Theorem (L¨anger,1980) F is associative and quasitrivial m ( max j ; if A 6= B; - A×B 9 - : F jA×B = 8A; B 2 X = ∼ π1jA×B or π2jA×B ; if A = B; 6 max - π1 or π2 π1 or max - π2 - Quasitrivial semigroups Theorem (L¨anger,1980) F is associative and quasitrivial m ( max j ; if A 6= B; - A×B 9 - : F jA×B = 8A; B 2 X = ∼ π1jA×B or π2jA×B ; if A = B; 6 max - π1 or π2 π1 or max - π2 - Quasitrivial semigroups 6 6 4 1 4 1 3 3 3 s§ s s s¤ 3 s s s s 2 4 2 s s s s 4 s s s s 1 2 1 s¦ s s s¥ 2 s s s s s s s s - s s s s - 1< 2 < 3 < 4 2∼ 4≺ 3 ≺ 1 Order-preservable operations Definition. F : X 2 ! X is said to be ≤-preserving for some total ordering ≤ on X if for any x; y; x 0; y 0 2 X such that x ≤ x 0 and y ≤ y 0, we have F (x; y) ≤ F (x 0; y 0) Definition. We say that F : X 2 ! X is order-preservable if it is ≤-preserving for some ≤ Q: Given an associative and quasitrivial F , is it order-preservable? Order-preservable operations Definition. F : X 2 ! X is said to be ≤-preserving for some total ordering ≤ on X if for any x; y; x 0; y 0 2 X such that x ≤ x 0 and y ≤ y 0, we have F (x; y) ≤ F (x 0; y 0) Definition. We say that F : X 2 ! X is order-preservable if it is ≤-preserving for some ≤ Q: Given an associative and quasitrivial F , is it order-preservable? Order-preservable operations Definition. F : X 2 ! X is said to be ≤-preserving for some total ordering ≤ on X if for any x; y; x 0; y 0 2 X such that x ≤ x 0 and y ≤ y 0, we have F (x; y) ≤ F (x 0; y 0) Definition. We say that F : X 2 ! X is order-preservable if it is ≤-preserving for some ≤ Q: Given an associative and quasitrivial F , is it order-preservable? Order-preservable operations 6 max - π1 or π (∗) 2 π1 or max π2 - - 2-quasilinearity : a ≺ b ∼ c ∼ d =) a; b; c; d are not pairwise distinct Theorem Assume the axiom of choice. F is associative, quasitrivial, and order-preservable m 9 - : F is of the form (∗) and - is 2-quasilinear Order-preservable operations 6 max - π1 or π (∗) 2 π1 or max π2 - - 2-quasilinearity : a ≺ b ∼ c ∼ d =) a; b; c; d are not pairwise distinct Theorem Assume the axiom of choice. F is associative, quasitrivial, and order-preservable m 9 - : F is of the form (∗) and - is 2-quasilinear Order-preservable operations 6 max - π1 or π (∗) 2 π1 or max π2 - - 2-quasilinearity : a ≺ b ∼ c ∼ d =) a; b; c; d are not pairwise distinct Theorem Assume the axiom of choice.