C-semiring: Difference between revisions
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#redirect [[Semiring]] |
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In [[abstract algebra]], a '''c-semiring''' (that is, a '''constraint-based [[semiring]]''') is a [[tuple]] <''A'',+,X,'''0''','''1'''> such that: |
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<ul> |
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<li>''A'' is a set and '''0''', '''1''' are elements of ''A''.</li> |
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<li>+ is the additive operation and is a [[commutative]] (i.e., +(''a'',''b'') = +(''b'',''a'')) and [[associative]] (i.e., +(''a'',+(''b'',''c'')) = +(+(''a'',''b''),''c'')) operation such that +(''a'','''0''') = ''a'' = +('''0''',''a'') (i.e., '''0''' is its unit element). |
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<li>+ is defined over (possibly infinite) sets of elements of ''A'' as follows: |
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<ul> |
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<li>for all ''a'' which are elements of ''A'', +({''a''}) = ''a''; </li> |
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<li>+(empty set) = '''0''' and +(''A'') = '''1''';</li> |
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<li>+(U''A<sub>i</sub>'', ''i'' element of ''S'') = +({+(''A<sub>i</sub>''), ''i'' element of ''S''}) for all sets of indices of ''S'' (flattening property).</li> |
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</ul></li> |
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<li>X is called the multiplicative operation, is a [[binary function|binary]], [[associative]] and [[commutative]] operation such that '''1''' is its unit element and ''a'' X '''0''' = '''0''' = '''0''' X ''a'' (i.e., '''0''' is its absorbing element); </li> |
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<li>X distributes over +.</li> |
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</ul> |
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A '''c-semiring''' is a [[semiring]] with idempotent addition. |
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{{DEFAULTSORT:C-Semiring}} |
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[[Category:Algebraic structures]] |
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[[Category:Ring theory]] |
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{{Abstract-algebra-stub}} |
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Latest revision as of 14:41, 8 June 2019
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