By Tomasz Brzezinski, José Luis Gomez-Pardo, Ivan Shestakov, Patrick F. Smith

The 23 articles during this quantity surround the complaints of the overseas convention on Modules and Comodules held in Porto (Portugal) in 2006 and devoted to Robert Wisbauer at the party of his sixty fifth birthday. those articles mirror Professor Wisbauer's extensive pursuits and provides an outline of other fields with regards to module conception, a few of that have an extended culture while others have emerged lately. They contain themes within the formal idea of modules bordering on class thought, in ring thought, in Hopf algebras and quantum teams, and in corings and comodules.

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**Example text**

3) The following subcategories are mutually equivalent: Vr (MT ) = Wr (MT ) = Xr (MT ) = Xr (MT ) ≈ Xr (MS ) = Xr (MS ) = Wr (MS ) = Vr (MS ). 4) If Vr (MT ) ≤ LI and Wr (MT ) ≤ CJ , then Vr (MT ) ≈ Wr (MS ). If Wr (MT ) ≤ CJ and Vr (MS ) ≤ LI , then Vr (MS ) ≈ Wr (MT ). Y. K. Nauman Proof. 5. If Vl (MT ) ≤ I L and Wl (MS ) ≤ J C, then we have Vl (MT ) = Vl (MT ) ∩ IL = Vl (MT ) ≈ Wl (MS ) = Wl (MS ) ∩ On the other hand, if Wl (MT ) ≤ I L and Vl (MS ) ≤ Wl (MT ) = Wl (MT ) ∩ IC J C, JC = Wl (MS ).

1. The following relations hold. τab ◦ ξtn = 0, for every t ≤ a ≤ b ≤ n. ◦ ξbn ◦ σba = IdXb /Xa , for every a ≤ b ≤ n. λab ◦ ξub = λau , for every a ≤ u ≤ b. ξun ◦ σuu−1 = δa,u IdXa /Xa−1 for every 0 ≤ a, u (7) τab a τa−1 ◦ ξat ◦ σaa−1 ◦ a ωa−1 ◦ λat (8) (9) ≤ n. = IdXt . (10) (11) 1≤a≤t ωtn = b ωtn ξbn σbb−1 τb−1 , for every 0 ≤ u ≤ t ≤ n. (12) ωjn ξin = 0, for every i ≤ j. (13) u+1≤b≤n σnj ωjn ξin σii−1 = ξin σii−1 , for every j + 1 ≤ i. (14) Proof. (7) follows by the following argument: (2) τab ◦ ξtn = ωab ◦ λbn ◦ ξtn = ωab ◦ λbn ◦ ξbn ◦ ξtb (1) = ωab ◦ ξtb = ωab ◦ ξab ◦ ξta (3) = 0.

Let u, v, w ∈ L be such that u = vw. Then, u = vw is the Shirshov decomposition of u if and only if either v ∈ X, or else if v = v1 v2 is the Shirshov decomposition of v, then w ≤ v2 . 2. Braided vector spaces of diagonal type We brieﬂy recall some notions we shall work with; we refer to [AS2] for more details. A braided vector space is a pair (V, c), where V is a vector space and c ∈ Aut(V ⊗V ) is a solution of the braid equation: (c ⊗ id)(id ⊗c)(c ⊗ id) = (id ⊗c)(c ⊗ id)(id ⊗c). We extend the braiding to c : T (V )⊗T (V ) → T (V )⊗T (V ) in the usual way.