### Table 3: Commutative division algebras.

"... In PAGE 32: ... Proposition 2.2 [1, Theorem 3] An algebra given by Table3 is a division algebra if and only if d2 lt; 4b a b c d . First, we consider the case when A has exactly one idempotent.... In PAGE 33: ... Lemma 2.4 An algebra determined by Table3 has exactly one idempotent if and only if either (2a d)2 lt; 4c(1 2b) or d = 2a; b = 1 2. Taking into account Proposition 2.... ..."

### Table 3: Commutative division algebras.

2004

Cited by 6

### Table 3: Commutative division algebras.

2004

Cited by 6

### Table 1. (Continued.) Algebra Non-zero commutation relations Invariants

"... In PAGE 8: ...V Boyko et al Table1 . (Continued.... ..."

### Table 1: Hopf algebras.

2002

"... In PAGE 26: ... A Hopf algebra is a k-vector space, H, equipped with an algebra structure, a com- patible coalgebra structure (= bialgebra) and an antipode satisfying the appropriate equations. Table1 summarizes the necessary structure [29]. We say a Hopf algebra is (co)commutative if the (co)multiplication is (co)commutative, i.... ..."

Cited by 2

### Table 2.3: Multiplication table of the eight{dimensional commutative algebra

### Table 1: Real four dimensional Lie algebras without commutative three dimen- sional subalgebra

### Table 1: Commutation relations for GA(2):

1999

"... In PAGE 4: ... This leads naturally to a local vector space representation for in nitesi- mal transformations, in which an a ne transforma- tion matrix, A can be obtained from a vector, A by the exponential map: A = ePi AiGi (9) where eX = I + X + 1 2X2 + 1 6X3 + : : : For small Ai this can be approximated by the linear term and the Gi form a basis for a vector space, known as a Lie Algebra. Formally, a Lie Algebra is a vector space together with a bilinear anti-symmetric opera- tor, the Lie Bracket, satisfying the Jacobi identity: [A; [B; C]] + [B; [C; A]] + [C; [A; B]] = 0 (10) Where a Lie Algebra is obtained from a group in the manner identi ed above, the Lie Bracket is de ned by the commutator of the generators: [A; B] = C (11) where C is de ned by X i;j AiBj(GiGj ? GjGi) = X k CkGk (12) The commutation relations for GA(2) are shown in Table1 . The in nitesimal representation of the Lie Al- gebra can be extended by considering the exponential map for nite transformations.... ..."

Cited by 19

### Table 1. Ore algebras

1996

"... In PAGE 4: ...elds. We specify commutative ring or commutative eld when necessary. Moreover, all rings under consideration in this paper are of characteristic 0. Table1 gives examples of the type of operators we consider. All these operators share a very simple commutation rule of the variable @ with polynomials in x.... In PAGE 5: ...3 Examples of skew polynomial rings are given in Table1 . In all the cases under consideration in this table, A is of either form K[x] or K(q)[x] with K a eld.... In PAGE 6: ...lgebra F of functions, power series, sequences, distributions, etc. Then Eq. (1) extends to the following Leibniz rule for products 8f; g 2 F @i(fg) = i(f)@i(g) + i(f)g: (6)This makes F an S-algebra. The actions of the operators corresponding to important Ore algebras are given in Table1 . In the remainder of this article, we use the word \function quot; to denote any object on which the elements of an Ore algebra act.... In PAGE 8: ... Then O is left Noetherian and a non-commutative version of Buchberger apos;s algorithm terminates. As can be seen from Table1 , this theorem implies that many useful Ore algebras are left Noe-... In PAGE 12: ... Then the annihilating ideal for any product fg where f is annihilated by I and g is annihilated by K is also @- nite. As can be seen from Table1 , this hypothesis does not represent a severe restriction on the class of Ore algebras we consider. Again, f and g in this lemma need not be interpreted as functions but as generators of the O-mod-... ..."

### Table 5. Laws for Commuting and Distributing Update Connectives

2006

"... In PAGE 55: ...Schema Variables Table5 . Modi ers for Schema Variables Modi er Applicable to rigid \term A \formula Terms or formulae that can syntactically be identi ed as rigid strict \term A Terms of type A (and not of proper subtypes of A) list \program t Sequences of program entities.... In PAGE 106: ...xample 2. We continue Example 1 and assume the same vocabulary/algebra. a := 1 ; f(a) := 2 a := 1 j f(1) := 2 valS; (a := 1 ; f(a) := 2) = fhai 7! 1; hf; (1)i 7! 2g valS; (a := 1 ; (a := 3 j f(a) := 2)) = fhai 7! 3; hf; (1)i 7! 2g We normalise the update in the second line using the given rewriting rules: a := 1 ; (a := 3 j f(a) := 2) (R45) ! a := 1 j fa := 1g (a := 3 j f(a) := 2) (R48) ! a := 1 j (fa := 1g a := 3 j fa := 1g f(a) := 2) (R47) ! a := 1 j (a := fa := 1g 3 j f(fa := 1g a) := fa := 1g 2) (R2); (R12) ! a := 1 j (a := 3 j f(non-rec(a := 1; a; ())) := 2) (R11) ! a := 1 j (a := 3 j f(if true then 1 else a) := 2) The last expression can be simpli ed further using rules for conditional terms, which are, however, beyond the scope of this paper. Further, using (R54) in Table5 , it is possible to eliminate the assignment a := 1, which is overridden by a := 3. 8 Soundness and Completeness of Update Application The following two lemmas state that the rewriting rules from Sect.... In PAGE 111: ...ewriting rules for update application (than the ones given in Sect. 5). This has been done for the implementation of updates in KeY. Table5 gives, besides others, identities that enable to establish form (1) by turning sequential composition into parallel composition, distributing if and for through parallel composition and commuting if and for. Another impor-... ..."