Daniel Simson left us unexpectedly on the 16th of April 2022.He served as editor of Fundamenta Informaticae since 2011.An eminent mathematician, he made a lasting contribution to modern algebra, in particular by his work on Grothendieck categories.Since the last two decades, Simson showed a vivid interest in mathematical challenges of computer science.His own work concentrated on symbolic algorithms issuing from algebra and spectral analysis of graphs, but he also animated a group of young mathematicians working in the area.His role in Fundamenta Informaticae was invaluable for his unlimited competence in mathematics, continuous readiness to help, and perfect manners.For several generations of Polish mathematicians, Professor Daniel Simson embodied the highest values of academic work.

This is the first part of our two part paper with the same title. Following our Coxeter spectral study in [Fund. Inform. [123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827-854] of the category 𝒰 ℬ i g r n of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study here the larger category ℛ ℬ i g r n of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual ℤ-congruences ~ Z and ≈ Z . The positive graphs Δ in ℛ ℬ i g r n , with dotted loops, are studied by means of the complex Coxeter spectrum s p e c c Δ ⊂ ℂ , the irreducible mesh root systems of Dynkin types 𝔹 n , n ≥ 2 , ℂ n , n ≥ 3 , 𝔽 4 , 𝔾 2 , the isotropy group Gl( n, ℤ) Δ (containing the Weyl group of Δ), and by applying the matrix morsification technique introduced in [J. Pure A ppl. Algebra 215(2011), 13-24] and [Fund. Inform. [123(2013), 447-490]. One of our aims of the paper is to study the Coxeter spectral analysis question: “ Does the congruence Δ ≈ ℤ Δ′ hold, for any pair of connected positive graphs Δ , Δ ′ ∈ ℛ ℬ i g r n such that spec c Δ = spec c Δ ′ and the numbers of loops in Δ and Δ′ coincide?” We do it by a reduction to the Coxeter spectral study of the G1 ( n , ℤ ) D -orbits in the set M o r D ⊂ 𝕄 n ( ℤ ) of matrix morsifications of a Dynkin diagram D = D Δ ∈ 𝒰 B i g r n

This is the second part of our two part paper with the same title. Following our Coxeter spectral study in [Fund. Inform. [123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827-854] of the category 𝒰 ℬ i g r n of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study here the larger category ℛ ℬ i g r n of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual ℤ-congruences ~ Z and ≈ Z . The positive graphs Δ in ℛ ℬ i g r n , with dotted loops, are studied by means of the complex Coxeter spectrum s p e c c Δ ⊂ ℂ , the irreducible mesh root systems of Dynkin types 𝔹 n , n ≥ 2 , ℂ n , n ≥ 3 , 𝔽 4 , 𝔾 2 , the isotropy group Gl( n, ℤ) Δ (containing the Weyl group of Δ), and by applying the matrix morsification technique introduced in [J. Pure A ppl. Algebra 215(2011), 13-24] and [Fund. Inform. [123(2013), 447-490]. One of our aims of our two part paper is to study the Coxeter spectral analysis question: “ Does the congruence Δ ≈ ℤ Δ′ hold, for any pair of connected positive graphs Δ , Δ ′ ∈ ℛ ℬ i g r n such that spec c Δ = spec c Δ ′ and the numbers of loops in Δ and Δ′ coincide?” We do it by a reduction to the Coxeter spectral study of the G1 ( n , ℤ ) D -orbits in the set M o r D ⊂ 𝕄 n ( ℤ ) of matrix morsifications of a Dynkin diagram D = D Δ ∈ 𝒰 B i g r

This paper can be viewed as a third part of our paper [Fund. Inform. 2015, in press]. Following our Coxeter spectral study in [Fund. Inform. 123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827-854] of the category 𝒰 ℬ i g r n of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study a larger category ℛ ℬ i g r n of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual ℤ-congruences ∼ Z and ≈ Z . The positive graphs Δ in ℛ ℬ i g r n , with dotted loops, are studied by means of the complex Coxeter spectrum s p e c c Δ ⊂ ℂ , the irreducible mesh root systems of Dynkin types 𝔹 n , n ≥ 2 , ℂ n , n ≥ 3 , 𝔽 4 , 𝔾 2 , the isotropy group Gl( n, ℤ) Δ (containing the Weyl group of Δ), and by applying the matrix morsification technique introduced in [J. Pure A ppl. Algebra 215(2011), 13-24] Here we present combinatorial algorithms for constructing the isotropy groups G1 ( n , ℤ ) Δ . One of the aims of our three paper series is to develop computational tools for the study of the ℤ-congruence ∼ ℤ and the following Coxeter spectral analysis question: “ Does the congruence Δ ≈ ℤ Δ′ holds, for any pair of connected positive graphs Δ , Δ ′ ∈ ℛ ℬ i g r n such that spec c Δ = spec c Δ ′ and the numbers of loops in Δ and Δ′ coincide?” For this purpose, we construct in this paper a extended inflation algorithm Δ ↦ 𝒟 Δ , with 𝒟 Δ ∼ ℤ Δ , that allows a reduction of the question to the Coxeter spectral study of the G1 ( n , ℤ )

Inspired by the bimodule matrix problem technique and various classification problems in poset representation theory, finite groups and algebras, we study the action of Belitskii algorithm on a class of square n by n block matrices M with coefficients in a field K. One of the main aims is to reduce M to its special canonical form M ∞ with respect to the conjugation by elementary transformations defined by a class of matrices chosen in a subalgebra of the full matrix algebra $\mathbb{M}_n$(K). The algorithm can be successfully applied in the study of indecomposable linear representations of finite posets by a computer search using numeric and symbolic computation. We mainly study the case when the di-graph (quiver) associated to the output matrix M ∞ of the algorithm is a disjoint union of trees. We show that exceptional representations of any finite poset are determined by tree matrices. This generalizes a theorem of C.M. Ringel proved for linear representations of di-graphs.