Homomorphic equivalence

The above type of equivalence between instances with nulls is captured, in general, by the notion of homomorphic equivalence. Recall that two instances 1 and 12 are homomorphically equivalent, with notation I /2, if there exist homomorphisms in both directions between I and I2. 

Intuitively, the above definition uses homomorphic equivalence as a replacement for the usual equality between instances. This is consistent with the fact that, in the presence of nulls, the notion of homomorphism itself becomes a replacement for the usual containment between instances. Note that when 7i and 72 are ground, h h is the same as 7t c /2. However, when 7] has nulls, these nulls are allowed... 

The existence of a chase-inverse for M implies that M has no information loss, since we can recover an instance that is the same modulo homomorphic equivalence as the original source instance. At the same time, a chase-inverse is a relaxation of the notion of an exact chase-inverse hence, it may exist even when an exact chase-inverse does not exist. 

As it can be seen, the recovered source instance U is not homomorphically equivalent to the original source instance there is a homomorphism from U to 7, but no homomorphism can map the constant CS in 7 to the null X in U. Intuitively, there is information loss in the evolution mapping M", which does not export the major field. Later on, in Sect. 5.2, we will show that in fact M" has no chase-inverse thus, we cannot recover a homomorphically equivalent source instance. 

Both possibilities are shown. Scheme P is a strong homomorphic image of each of the two resulting schemes all three schemes are strongly equivalent. 

Every program scheme P can be converted into a strongly equivalent block program scheme P with the same set of variables, such that P is a graph homomorphic image of P. ... 

There are two possible approaches. In one approach we in effect build the execution sequence tree for P. We start with node (1,0) labelled START. A node (k,r) will be at level r of the new tree-like structure and be labelled with the instruction named by k. Suppose statement k in P is connected by an arrow (with or without a label) to statement p in P and that we have constructed node (k,r) in P to date. If (k,r) has no ancestor of the form (p,r ), r < r, place node (p,r+l) labelled by statement p on the tree, with an arrow from (k,r) to (p,r+l) which contains any label on the arrow from k to p. If there is already an ancestor (p,rT), r < r, of (k,r) on the tree, then do not create (p,r+l) but instead add an arrow from (k,r) back to (p,r ) containing any label also on the arrow from k to p. If P has N statements, this process must terminate in a scheme P with at most N levels. Clearly P is tree-like and is strongly equivalent to P. This transformation is global and structure preserving. In fact P is a strong homomorphic image of P under the homomorphism h taking each (k,r>) back into k. ... 

What familiar condition on the quaternion u + xi + yj + "k w equivalent to requiring the corresponding matrix to be an element o/ 5(9(1, 3) Use this calculation to define a group homomorphism from the set of quaternions satisfying that condition to SO (A). 

There are several different methods of obtaining sets of matrices which are homomorphic with a given point group and in this chapter we discuss these methods in some detail. One way is to consider the effect that a symmetry operation has on the Cartesian coordinates of some point (or, equivalently, on some position vector) in the molecule. Another way is to consider the effect that a symmetry operation has on one or more sets of base vectors (coordinate axes) within the molecule. 

We keep the notation of the proof of (41). The hypothesis of the corollary implies that the surjective homomorphism p. PAM is already given. Hence the flatness of PA/ is equivalent to the existence of the surjection q pA - R, and this is another way to formulate what has to be proved. 

The term equivalent is overly general and therefore bland and of equivocal meaning. Thus the methylene hydrogen atoms in propionic acid (Fig. 1) are equivalent when detached (i.e. they are homomorphic), but, as already explained, they are not equivalent in the CH3CH2C02H molecules because of their placement — i.e. they are heterotopic. Ligands that are equivalent by the criteria to be described in the sequel are called homotopic from Greek homos = same and topos = place 6>, those that are not are called heterotopic . 

One can verify directly that this is a homomorphism GLj - GL3. Obviously it contains information specifically about quadratic forms as well as about GL2—the orbits are isometry classes. We will touch on this again when we mention invariant theory in (16.4), but for now we use representations merely as a tool for deriving structural information about group schemes. The first step is to use a Yoneda-type argument to find the Hopf-algebra equivalent. 

Clearly we can define these concepts using just the T-action. Let T be any group acting as automorphisms of a group F. The maps / T - F satisfying f at) =/(crossed homomorphisms (they are homomorphisms when the action is trivial). The ones cohomologous to/are those of the form cf (o)[a(c)Y1 for some fixed c in F this is the definition that matches up with ours for G(S), and one can easily check that it is an equivalence relation in general. The set of equivalence classes is denoted // (T, F). 

In sum, homomorphic ligands (i.e., those which are identical when detached from the molecule) can be classified as homotopic (equivalent) or heterotopic (nonequivalent). The latter can be further distinguished as diastereotopic or enantiotopic. These divisions are shown in Figure A7-5, and the descriptions presented for the different types of ligands are collected in Table A7-1. 

Putting it all together, we now formally capture the two desiderata discussed above (data exchange equivalence and homomorphic containment) into the following definition of a relaxed chase-inverse. 

The notion of relaxed chase-inverse originated in Fagin et al. [2009b], under the name of universal-faithful inverse. The definition given in Fagin et al. [2009b] had, however, a third condition called universality, which turned out to be redundant (and equivalent to homomorphic containment). Thus, the formulation given here for a relaxed chase-inverse is simpler. 

Notice that these stalks are local rings (since the stalks on affine schemes are always local rings). Then condition (b) asserts that f is a local homomorphism, i.e., equivalently... 

A faciononselective reaction is one in which (a) every pathway (transition state) involved in the reaction at face 1 has an isoenergetic counterpart at face 2 (the corresponding transition state pairs are either homomorphic or enantiomorphic), and (b) at least one isoenergetic pair of enantiomorphic transition states is involved (the other(s) being homomorphic). Faciononselective processes are represented by quartets q3-q21. In real terms, the product(s) from face 1 will be formed in amounts identical with that(those) from face 2 if one of the products (or the only product) from face 1 is chiral, then an equivalent amount of the enantiomeric counterpart win result from face 2. 

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