### 7.7.4 References and history of LETTERPLACE

LETTERPLACE has undergone several stages of development.

The first one, the pure Letterplace implementation for homogeneous ideals, was created by V. Levandovskyy and H. Schoenemann in 2007-2009.

Later in 2010-2014, experiments with advanced (among other, with shift-invariant) data structures were performed by V. Levandovskyy, B. Schnitzler and G. Studzinski, and new libraries for -dimension, -bases, and Ufnarovskij graph were written.

The next stage started in 2017, when K. Abou Zeid joined the team of H. Schoenemann and V. Levandovskyy. Those recent activities led to the change of interface to the one, usual in the free algebra. The Letterplace data structure is still at heart of the implementation, though not explicitly visible by default. It has been generalized to support as coefficient ring (together with T. Metzlaff (RWTH Aachen and INRIA Sophia Antipolis)); to support bimodules and compute syzygies and lifts, to name a few. We are grateful to L. Schmitz (RWTH Aachen) for his contributions to the development.

References:

[LL09]: Roberto La Scala and Viktor Levandovskyy, "Letterplace ideals and non-commutative Groebner bases", Journal of Symbolic Computation, Volume 44, Issue 10, October 2009, Pages 1374-1393, see http://dx.doi.org/10.1016/j.jsc.2009.03.002.

[LL13]: Roberto La Scala and Viktor Levandovskyy, "Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra", Journal of Symbolic Computation, Volume 48, Issue 1, January 2013, Pages 1374-1393, see http://dx.doi.org/10.1016/j.jsc.2012.05.003 and also http://arxiv.org/abs/1009.4152.

[LSS13]: Viktor Levandovskyy, Grischa Studzinski and Benjamin Schnitzler , "Enhanced Computations of Groebner Bases in Free Algebras as a New Application of the Letterplace Paradigm", Proc. ISSAC 2013, ACM Press, 259-266, see https://doi.org/10.1145/2465506.2465948.

[L14]: Roberto La Scala, "Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms", International Journal of Algebra and Computation, Volume 24, Number 08, Pages 1157-1182, 2014, see also https://doi.org/10.1142/S0218196714500519.

[Mora16]: Teo Mora, "Solving Polynomial Equation Systems IV: Volume 4, Buchberger Theory and Beyond.", Cambridge University Press, 2016.

[LMZ20]: Viktor Levandovskyy, Tobias Metzlaff and Karim Abou Zeid, "Computation of free non-commutative Groebner Bases over with SINGULAR:LETTERPLACE", Proc. ISSAC 2020, Pages 312-319, ACM Press (2020), https://dl.acm.org/doi/10.1145/3373207.3404052. Video of the talk is at https://av.tib.eu/media/50124.

[LSZ20]: Viktor Levandovskyy, Hans Schoenemann and Karim Abou Zeid, "LETTERPLACE - a Subsystem of SINGULAR for computations with free algebras via Letterplace Embedding", Proc. ISSAC 2020, 305-311, ACM Press, https://dl.acm.org/doi/10.1145/3373207.3404056. Video of the talk is at https://av.tib.eu/media/50123.

[SL20]: Leonard Schmitz and Viktor Levandovskyy : Formally Verifying Proofs for Algebraic Identities of Matrices . In: Intelligent Computer Mathematics (Proceedings of the CICM 2020), Pages 222-236, Springer LNAI, LNCS (2020).

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