Tropical matrix algebra
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1 Tropical matrix algebra Marianne Johnson 1 & Mark Kambites 2 NBSAN, University of York, 25th November University of Manchester. Supported by CICADA (EPSRC grant EP/E050441/1). 2 University of Manchester. Supported by an RCUK Academic Fellowship. Johnson & Kambites Tropical matrix algebra 1 / 17
2 The tropical semiring The tropical (or max-plus) semiring has elements R = R { } and binary operations x y = max(x, y); and x y = x + y. Johnson & Kambites Tropical matrix algebra 2 / 17
3 The tropical semiring The tropical (or max-plus) semiring has elements R = R { } and binary operations x y = max(x, y); and x y = x + y. Properties R is an idempotent semifield: Johnson & Kambites Tropical matrix algebra 2 / 17
4 The tropical semiring The tropical (or max-plus) semiring has elements R = R { } and binary operations x y = max(x, y); and x y = x + y. Properties R is an idempotent semifield: (R, ) is an abelian group with identity 0; is a zero element for ; (R, ) is a commutative monoid with identity ; distributes over ; x x = x Johnson & Kambites Tropical matrix algebra 2 / 17
5 The tropical semiring Tropical matrix algebra or max-plus algebra is linear algebra where the base field is replaced by the tropical semiring. Johnson & Kambites Tropical matrix algebra 3 / 17
6 The tropical semiring Tropical matrix algebra or max-plus algebra is linear algebra where the base field is replaced by the tropical semiring. Applications Tropical methods have applications in... Johnson & Kambites Tropical matrix algebra 3 / 17
7 The tropical semiring Tropical matrix algebra or max-plus algebra is linear algebra where the base field is replaced by the tropical semiring. Applications Tropical methods have applications in... Combinatorial Optimisation Discrete Event Systems Control Theory Formal Languages and Automata Phylogenetics Statistical Inference Geometric Group Theory Enumerative Algebraic Geometry Johnson & Kambites Tropical matrix algebra 3 / 17
8 Tropical matrices We (hope to) study the semigroup M n (R) of all n n tropical matrices under multiplication. Johnson & Kambites Tropical matrix algebra 4 / 17
9 Tropical matrices We (hope to) study the semigroup M n (R) of all n n tropical matrices under multiplication. Question What is its abstract algebraic structure? Johnson & Kambites Tropical matrix algebra 4 / 17
10 Tropical matrices We (hope to) study the semigroup M n (R) of all n n tropical matrices under multiplication. Question What is its abstract algebraic structure? For example, what are its... Ideals? Idempotents? Subgroups? Johnson & Kambites Tropical matrix algebra 4 / 17
11 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Johnson & Kambites Tropical matrix algebra 5 / 17
12 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... Johnson & Kambites Tropical matrix algebra 5 / 17
13 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... (-,b) (a,b) (c,d) (-,- ) (c,- ) Johnson & Kambites Tropical matrix algebra 5 / 17
14 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... (-,b) (a,b) (c,b) = (a,b) + (c,d) (c,d) (-,- ) (c,- ) Johnson & Kambites Tropical matrix algebra 5 / 17
15 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... (-,b) (a,b) (c,b) = (a,b) + (c,d) k x (c,b) (c,d) (-,- ) (c,- ) Johnson & Kambites Tropical matrix algebra 5 / 17
16 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Johnson & Kambites Tropical matrix algebra 6 / 17
17 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Example (-,f) m x (a,b) (a,b) (c,d) k x (c,d) (-,- ) (e,- ) Johnson & Kambites Tropical matrix algebra 6 / 17
18 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Example (a,b) (-,f) [-,f ] m x (a,b) (c,d) [a,b] [c,d] k x (c,d) (-,- ) (e,- ) [e,- ] Johnson & Kambites Tropical matrix algebra 6 / 17
19 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Example (a,b) (-,f) [-,f ] m x (a,b) (c,d) [a,b] [c,d] b-a k x (c,d) d-c (-,- ) (e,- ) [e,- ] - Johnson & Kambites Tropical matrix algebra 6 / 17
20 Projective tropical 1-space Thus we identify projective tropical 1-space with the two-point compactification of the real line ˆR = R {, } via the map [-,f ] [x, y] y x. [a,b] [c,d] b-a d-c [e,- ] - Johnson & Kambites Tropical matrix algebra 7 / 17
21 Projective tropical 1-space Thus we identify projective tropical 1-space with the two-point compactification of the real line ˆR = R {, } via the map [-,f ] [x, y] y x. [a,b] [c,d] b-a d-c [e,- ] - Question How does the algebraic structure of M n (R) relate to the geometric structure of affine tropical n-space and projective tropical (n 1)-space? Johnson & Kambites Tropical matrix algebra 7 / 17
22 Column and row spaces For A M n (R) we write C(A) for the column span of A (a tropical subspace in R n ); R(A) for the row span of A (a tropical subspace in R n ). Johnson & Kambites Tropical matrix algebra 8 / 17
23 Column and row spaces For A M n (R) we write C(A) for the column span of A (a tropical subspace in R n ); R(A) for the row span of A (a tropical subspace in R n ). Example ( ) a Let A = M b c 2 (R), where a, b, c R. Johnson & Kambites Tropical matrix algebra 8 / 17
24 Column and row spaces For A M n (R) we write C(A) for the column span of A (a tropical subspace in R n ); R(A) for the row span of A (a tropical subspace in R n ). Example ( ) a Let A = M b c 2 (R), where a, b, c R. Then C(A) = {( x y ) } : x + b a y R 2. b-a Johnson & Kambites Tropical matrix algebra 8 / 17
25 Projective column and row spaces For A M n (R) we write PC(A) for the image of C(A) in projective space (a convex set); PR(A) for the image of R(A) in projective space (a convex set). Johnson & Kambites Tropical matrix algebra 9 / 17
26 Projective column and row spaces For A M n (R) we write PC(A) for the image of C(A) in projective space (a convex set); PR(A) for the image of R(A) in projective space (a convex set). Example In the case n = ( 2, convex ) sets in ˆR are just intervals. a Consider A = M b c 2 (R), where a, b, c R. Johnson & Kambites Tropical matrix algebra 9 / 17
27 Projective column and row spaces For A M n (R) we write PC(A) for the image of C(A) in projective space (a convex set); PR(A) for the image of R(A) in projective space (a convex set). Example In the case n = ( 2, convex ) sets in ˆR are just intervals. a Consider A = M b c 2 (R), where a, b, c R. b-a Then PC(A) = [b a, ] ˆR. Johnson & Kambites Tropical matrix algebra 9 / 17
28 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. Johnson & Kambites Tropical matrix algebra 10 / 17
29 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation xry xm = ym x R y and y R x Johnson & Kambites Tropical matrix algebra 10 / 17
30 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation Similarly... xry xm = ym x R y and y R x x L y Mx My, x J y MxM MyM, xly Mx = My xj y MxM = MyM; Johnson & Kambites Tropical matrix algebra 10 / 17
31 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation Similarly... xry xm = ym x R y and y R x x L y Mx My, x J y MxM MyM, We also define equivalence relations... xhy xry and xly; xly Mx = My xdy xrz and zly for some z M; xj y MxM = MyM; Johnson & Kambites Tropical matrix algebra 10 / 17
32 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation Similarly... xry xm = ym x R y and y R x x L y Mx My, x J y MxM MyM, We also define equivalence relations... Note xhy xry and xly; xly Mx = My xdy xrz and zly for some z M; xj y MxM = MyM; These relations encapsulate the (left, right and two-sided) ideal structure of M and are fundamental to its structure. Johnson & Kambites Tropical matrix algebra 10 / 17
33 Green s R relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A R B; (ii) C(A) C(B); (iii) PC(A) PC(B). Johnson & Kambites Tropical matrix algebra 11 / 17
34 Green s R relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A R B; (ii) C(A) C(B); (iii) PC(A) PC(B). Corollary Let A, B M n (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). Johnson & Kambites Tropical matrix algebra 11 / 17
35 Green s R relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A R B; (ii) C(A) C(B); (iii) PC(A) PC(B). Corollary Let A, B M n (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). So R-classes in M n (R) are in 1-1 correspondence with n-generated convex sets in projective tropical (n 1)-space. Johnson & Kambites Tropical matrix algebra 11 / 17
36 Green s L relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A L B; (ii) R(A) R(B); (iii) PR(A) PR(B). Corollary Let A, B M n (R). Then the following are equivalent: (i) ALB; (ii) R(A) = R(B); (iii) PR(A) = PR(B). So L-classes in M n (R) are in 1-1 correspondence with n-generated convex sets in projective tropical (n 1)-space. Johnson & Kambites Tropical matrix algebra 12 / 17
37 Green s R relation in M 2 (R). Corollary Let A, B M 2 (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). Johnson & Kambites Tropical matrix algebra 13 / 17
38 Green s R relation in M 2 (R). Corollary Let A, B M 2 (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). Corollary The lattice of principal right ideals in M 2 (R) is isomorphic to the intersection lattice generated by closed subintervals of the closed unit interval. Johnson & Kambites Tropical matrix algebra 13 / 17
39 Isometries in projective tropical 1-space We can define a metric on ˆR = R {, } by 0 if x = y d(x, y) = if x = y or x = y y x otherwise. This gives a natural notion of isometry (denoted by =). Johnson & Kambites Tropical matrix algebra 14 / 17
40 Isometries in projective tropical 1-space We can define a metric on ˆR = R {, } by 0 if x = y d(x, y) = if x = y or x = y y x otherwise. This gives a natural notion of isometry (denoted by =). Proposition Let A M 2 (R). Then PC(A) = PR(A). Johnson & Kambites Tropical matrix algebra 14 / 17
41 Green s J relation in M 2 (R) Proposition Let A, B M 2 (R). Then A J B if and only if PC(A) embeds isometrically in PC(B). Johnson & Kambites Tropical matrix algebra 15 / 17
42 Green s J relation in M 2 (R) Proposition Let A, B M 2 (R). Then A J B if and only if PC(A) embeds isometrically in PC(B). Theorem Let A, B M 2 (R). Then the following are equivalent (i) AJ B; (ii) ADB; (iii) PC(A) = PC(B) (iv) PR(A) = PR(B) Johnson & Kambites Tropical matrix algebra 15 / 17
43 Green s J relation in M 2 (R) Proposition Let A, B M 2 (R). Then A J B if and only if PC(A) embeds isometrically in PC(B). Theorem Let A, B M 2 (R). Then the following are equivalent (i) AJ B; (ii) ADB; (iii) PC(A) = PC(B) (iv) PR(A) = PR(B) Corollary The lattice of principal two-sided ideals in M 2 (R) is isomorphic to the lattice of isometry types of closed convex subsets of ˆR. Johnson & Kambites Tropical matrix algebra 15 / 17
44 Idempotents and regularity The idempotents in M 2 (R) are ( ) ( ) 0 x 0 x,, y x + y y 0 ( x + y x y 0 ) and ( ) where x, y R with x + y 0. Johnson & Kambites Tropical matrix algebra 16 / 17
45 Idempotents and regularity The idempotents in M 2 (R) are ( ) ( ) 0 x 0 x,, y x + y y 0 where x, y R with x + y 0. Fact ( x + y x y 0 ) and ( For every 2-generated convex subset X of ˆR, there is an idempotent E M 2 (R) with PC(E) = X. Thus M 2 (R) is regular. ) Johnson & Kambites Tropical matrix algebra 16 / 17
46 Idempotents and regularity The idempotents in M 2 (R) are ( ) ( ) 0 x 0 x,, y x + y y 0 where x, y R with x + y 0. Fact ( x + y x y 0 ) and ( For every 2-generated convex subset X of ˆR, there is an idempotent E M 2 (R) with PC(E) = X. Thus M 2 (R) is regular. ) Example Consider X = [b a, ] ˆR. Then ( we can choose ) 0 E = M b a 0 2 (R) such that PC(E) = X b-a Johnson & Kambites Tropical matrix algebra 16 / 17
47 Groups of 2 2 tropical matrices Let S be a semigroup. It is well known that the maximal subgroups of S are exactly the H-classes of idempotents and that any two maximal subgroups in the same D-class are isomorphic. Johnson & Kambites Tropical matrix algebra 17 / 17
48 Groups of 2 2 tropical matrices Let S be a semigroup. It is well known that the maximal subgroups of S are exactly the H-classes of idempotents and that any two maximal subgroups in the same D-class are isomorphic. Theorem Let M ˆR be a closed convex subset. The maximal subgroups in the D-class corresponding to M are isomorphic to: {1} if M = ; R if M is a point or an interval with one real endpoint; R S 2 if M is an interval with 2 real endpoints; R S 2 if M = ˆR. Johnson & Kambites Tropical matrix algebra 17 / 17
49 Groups of 2 2 tropical matrices Let S be a semigroup. It is well known that the maximal subgroups of S are exactly the H-classes of idempotents and that any two maximal subgroups in the same D-class are isomorphic. Theorem Let M ˆR be a closed convex subset. The maximal subgroups in the D-class corresponding to M are isomorphic to: Corollary {1} if M = ; R if M is a point or an interval with one real endpoint; R S 2 if M is an interval with 2 real endpoints; R S 2 if M = ˆR. Every group of 2 2 tropical matrices is torsion-free abelian, or has a torsion-free abelian subgroup of index 2. Johnson & Kambites Tropical matrix algebra 17 / 17
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