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By Eric Jespers

In the final decade, semigroup theoretical equipment have happened obviously in lots of points of ring thought, algebraic combinatorics, illustration idea and their functions. particularly, prompted through noncommutative geometry and the speculation of quantum teams, there's a starting to be curiosity within the category of semigroup algebras and their deformations.This paintings provides a accomplished remedy of the most effects and techniques of the idea of Noetherian semigroup algebras. those common effects are then utilized and illustrated within the context of vital periods of algebras that come up in various parts and feature been lately intensively studied. numerous concrete structures are defined in complete aspect, specifically interesting periods of quadratic algebras and algebras regarding workforce earrings of polycyclic-by-finite teams. those provide new periods of Noetherian algebras of small Gelfand-Kirillov measurement. the point of interest is at the interaction among their combinatorics and the algebraic constitution. This yields a wealthy source of examples which are of curiosity not just for the noncommutative ring theorists, but in addition for researchers in semigroup conception and sure elements of staff and team ring idea. Mathematical physicists will locate this paintings of curiosity because of the eye given to purposes to the Yang-Baxter equation.

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In the final decade, semigroup theoretical equipment have happened evidently in lots of elements of ring conception, algebraic combinatorics, illustration concept and their purposes. specifically, stimulated by way of noncommutative geometry and the speculation of quantum teams, there's a growing to be curiosity within the type of semigroup algebras and their deformations.

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Bk } represented by a set of Boolean expressions {b1 , . . , bk } and a block B = b ∈ Π, SmtRefine returns a partition ΠB of B given by mutually unsatisfiable Boolean expressions {b1 , . . , bm } such that for all s1 , s2 ∈ B: P(s1 , T ) = P(s2 , T ) for all T ∈ S/Π iff s1 |= bi ⇔ s2 |= bi for all 1 ≤ i ≤ m Example 8. Let PEx be the probabilistic program in Fig. 1. Furthermore, let the initial partition be Πinit = { pc = 4 , pc = 4 } given by the Boolean expressions b1 = (pc = 4) and b2 = (pc = 4).

In fact, while Sign is complete for (| x > 0 |), its complete shell Int for (| x := x ± 1 |) loses this completeness: for example, we have that Int((| x > 0 |){0, 2}) = [2, 2] while (| x > 0 |)Int Int({0, 2}) = [1, +∞) [0, 2] = [1, 2]. It is simple to observe that if we try to compensate this lack by the complete shell of Int for (| x > k |), for all k ∈ Z, we end up with the whole concrete domain ℘(Z): this depends on the fact that max{S ∈ ℘(Z) | (| x > k |)S ⊆ Z≥k+2 } = Z=k+1 and then by closing under intersections the family of sets {Z=k }k∈Z we get ℘(Z).

Sci. 26(1-2), 53–82 (1983) 5. : Model checking. The MIT Press (1999) 6. : M´ethodes it´eratives de construction et d’approximation de points fixes d’op´erateurs monotones sur un treillis, analyse s´emantique des programmes. D. dissertation, Universit´e Scientifique et M´edicale de Grenoble, Grenoble, France (1978) 7. : Constructive design of a hierarchy of semantics of a transition system by abstract interpretation. Theoretical Computer Science 277(1-2), 47–103 (2002) 8. : Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints.

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