Resolutions

Recall that a resolution of a block design is a partition of the blocks into subsets, each of which forms a partition of the point set. Such a partition of the block (multi)set can be represented as a function on the set of indices of blocks (the parts of the partition being the preimages of the elements in the range of the function). We thus store a resolution as a function_on_indices with domain="blocks".

An automorphism of a resolution is a permutation of the set of points of the design such that, if this permutation is applied to the elements of each block in each resolution class, the (multi)set of resolution classes is the same as before. The collection of all automorphisms of a resolution of a design forms a subgroup of the automorphism group of the design itself, and we use the same automorphism_group structure for the automorphism group of a resolution as we do for the automorphism group of a block design (although the automorphism_group_properties for a resolution are different than those for a block design).

We specify a resolution as follows:

resolution = element resolution {
    function_on_indices,
    resolution_automorphism_group ?
}

A block design $D$ may have more than one resolution. We say that two resolutions $R$ and $S$ of $D$ are isomorphic if there is an element $g$ in the automorphism group of $D$, such that, when $g$ is applied to the elements of each block in each resolution class of $R$, the resulting resolution is equal to $S$. Isomorphism defines an equivalence relation on the set of resolutions of $D$.

We use the element resolutions to store a nonempty list of (distinct) resolutions of a resolvable design. The attributes of this tag are used to specify whether the listed resolutions are pairwise nonisomorphic and whether all isomorphism classes of resolutions are represented in the list.

resolutions = element resolutions {
    attribute pairwise_nonisomorphic  { "true" | "false" | "unknown" } ,
    attribute all_classes_represented { "true" | "false" | "unknown" } ,
    resolution +
}

We now display a famous resolvable design, the affine plane of order 3, which has just one resolution.

<list_of_designs design_type="block_design" dtrs_protocol="1.1" no_designs="1"
 pairwise_nonisomorphic="true" xmlns="http://designtheory.org/xml-namespace">
    <block_design b="12" id="t2-v9-k3-L1-1" v="9">
        <blocks ordered="true">
            <block><z>0</z><z>1</z><z>2</z></block>
            <block><z>0</z><z>3</z><z>4</z></block>
            <block><z>0</z><z>5</z><z>6</z></block>
            <block><z>0</z><z>7</z><z>8</z></block>
            <block><z>1</z><z>3</z><z>5</z></block>
            <block><z>1</z><z>4</z><z>7</z></block>
            <block><z>1</z><z>6</z><z>8</z></block>
            <block><z>2</z><z>3</z><z>8</z></block>
            <block><z>2</z><z>4</z><z>6</z></block>
            <block><z>2</z><z>5</z><z>7</z></block>
            <block><z>3</z><z>6</z><z>7</z></block>
            <block><z>4</z><z>5</z><z>8</z></block>
        </blocks>
        <resolutions all_classes_represented="true"
         pairwise_nonisomorphic="true">
            <resolution>
                <function_on_indices domain="blocks" n="12" ordered="true"
                 title="resolution">
                    <map>
                        <preimage><z>0</z><z>10</z><z>11</z></preimage>
                        <image><z>0</z></image>
                    </map>
                    <map>
                        <preimage><z>1</z><z>6</z><z>9</z></preimage>
                        <image><z>1</z></image>
                    </map>
                    <map>
                        <preimage><z>2</z><z>5</z><z>7</z></preimage>
                        <image><z>2</z></image>
                    </map>
                    <map>
                        <preimage><z>3</z><z>4</z><z>8</z></preimage>
                        <image><z>3</z></image>
                    </map>
                </function_on_indices>
            </resolution>
        </resolutions>
    </block_design>
</list_of_designs>