The independence assumption for a series or parallel system when component lifetimes are exponential : John P. Klein and M. L. Moeschberger. IEEE Trans. Reliab.R-35 (3), 330 (1986)

A note on set-theoretic solutions of the Yang–Baxter equation

For a finite involutive non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation it is known that the structure monoid M ( X , r ) M(X,r) is a monoid of I-type, and the structure algebra K [ M ( X , r ) ] K[M(X,r)] over a field K K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M ( X , r ) M(X,r) and the algebra K [ M ( X , r ) ] K[M(X,r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M ( X , r ) M(X,r) as a regular submonoid in the semidirect product A ( X , r ) ⋊ Sym ⁡ ( X ) A(X,r)\rtimes \operatorname {Sym} (X) , where A ( X , r ) A(X,r) is the structure monoid of the rack solution associated to ( X , r ) (X,r) , we prove that K [ M ( X , r ) ] K[M(X,r)] is a finite module over a central affine subalgebra. In particular, K [ M ( X , r ) ] K[M(X,r)] is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by | X | |X| . We also characterize, in ring-theoretical terms of K [ M ( X , r ) ] K[M(X,r)] , when ( X , r ) (X,r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M ( X , r ) M(X,r) . These results allow us to control the prime spectrum of the algebra K [ M ( X , r ) ] K[M(X,r)] and to describe the Jacobson radical and prime radical of K [ M ( X , r ) ] K[M(X,r)] . Finally, we give a matrix-type representation of the algebra K [ M ( X , r ) ] / P K[M(X,r)]/P for each prime ideal P P of K [ M ( X , r ) ] K[M(X,r)] . As a consequence, we show that if K [ M ( X , r ) ] K[M(X,r)] is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, G 1 , … , G m G_1,\dotsc ,G_m , each being the group of quotients of a cancellative subsemigroup of M ( X , r ) M(X,r) such that the algebra K [ M ( X , r ) ] K[M(X,r)] embeds into M v 1 ⁡ ( K [ G 1 ] ) × ⋯ × M v m ⁡ ( K [ G m ] ) \operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m]) , a direct product of matrix algebras.

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation

To every involutive non-degenerate set-theoretic solution [Formula: see text] of the Yang–Baxter equation on a finite set [Formula: see text] there is a naturally associated finite solvable permutation group [Formula: see text] acting on [Formula: see text]. We prove that every primitive permutation group of this type is of prime order [Formula: see text]. Moreover, [Formula: see text] is then a so-called permutation solution determined by a cycle of length [Formula: see text]. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.

Set-theoretic solutions of the Yang–Baxter equation, braces and symmetric groups

Given a skew left brace [Formula: see text], a method is given to construct all the non-degenerate set-theoretic solutions [Formula: see text] of the Yang–Baxter equation such that the associated permutation group [Formula: see text] is isomorphic, as a skew left brace, to [Formula: see text]. This method depends entirely on the brace structure of [Formula: see text]. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation are discussed and many consequences are derived. In particular, for each positive integer n a finite square-free multipermutation solution of the Yang–Baxter equation with multipermutation level n and an abelian involutive Yang–Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is proved that finite non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation whose associated involutive Yang–Baxter group is abelian are multipermutation solutions. Earlier the authors proved this with the additional square-free hypothesis on the solutions. It is also proved that finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace are multipermutation solutions.

On the Yang–Baxter equation and left nilpotent left braces

We introduce the notion of non-degenerate solution of the braid equation on the incidence coalgebra of a locally finite order. Each one of these solutions induces by restriction a non-degenerate set-theoretic solution over the underlying set. So, it makes sense to ask if a non-degenerate set-theoretic solution on the underlying set of a locally finite order extends to a non-degenerate solution on its incidence coalgebra. In this paper we begin the study of this problem.

It is proved that if a left brace [Formula: see text] has the operation ∗ associative, then [Formula: see text] is a two-sided brace. Consequently, [Formula: see text] is a Jacobson radical ring. This answers a question of Cedó, Gateva-Ivanova and Smoktunowicz.

Solutions of the Yang–Baxter equation associated with a left brace

Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to $B$. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to construct new families of finite simple left braces.

Skew left braces and 2-reductive solutions of the Yang-Baxter equation

We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic).

In this paper, we consider series systems and parallel systems with the dependence between the component lifetimes modelled by an Archimedean copulas. We obtain sufficient and necessary conditions of relative ageing orders between series (parallel) systems with different component numbers, which partially generalize some main results of Misra and Francis. When the component lifetimes follow the scale model, we also characterize the ordering properties between the series systems and (n–1)-out-of-n systems (parallel systems and 2-out-of-n systems) by mixture distribution.