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<html><head><meta charset="utf-8"><title>Algebra.Morphism</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Morphisms between algebraic structures</a>
<a id="148" class="Comment">------------------------------------------------------------------------</a>
<a id="222" class="Symbol">{-#</a> <a id="226" class="Keyword">OPTIONS</a> <a id="234" class="Pragma">--cubical-compatible</a> <a id="255" class="Pragma">--safe</a> <a id="262" class="Symbol">#-}</a>
<a id="267" class="Keyword">module</a> <a id="274" href="Algebra.Morphism.html" class="Module">Algebra.Morphism</a> <a id="291" class="Keyword">where</a>
<a id="298" class="Keyword">import</a> <a id="305" href="Algebra.Morphism.Definitions.html" class="Module">Algebra.Morphism.Definitions</a> <a id="334" class="Symbol">as</a> <a id="337" class="Module">MorphismDefinitions</a>
<a id="357" class="Keyword">open</a> <a id="362" class="Keyword">import</a> <a id="369" href="Algebra.html" class="Module">Algebra</a>
<a id="377" class="Keyword">import</a> <a id="384" href="Algebra.Properties.Group.html" class="Module">Algebra.Properties.Group</a> <a id="409" class="Symbol">as</a> <a id="412" class="Module">GroupP</a>
<a id="419" class="Keyword">open</a> <a id="424" class="Keyword">import</a> <a id="431" href="Function.html" class="Module">Function</a> <a id="440" class="Keyword">hiding</a> <a id="447" class="Symbol">(</a><a id="448" href="Function.Core.html#655" class="Function">Morphism</a><a id="456" class="Symbol">)</a>
<a id="458" class="Keyword">open</a> <a id="463" class="Keyword">import</a> <a id="470" href="Level.html" class="Module">Level</a>
<a id="476" class="Keyword">open</a> <a id="481" class="Keyword">import</a> <a id="488" href="Relation.Binary.html" class="Module">Relation.Binary</a>
<a id="504" class="Keyword">import</a> <a id="511" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="544" class="Symbol">as</a> <a id="547" class="Module">EqR</a>
<a id="552" class="Keyword">private</a>
<a id="562" class="Keyword">variable</a>
<a id="575" href="Algebra.Morphism.html#575" class="Generalizable">a</a> <a id="577" href="Algebra.Morphism.html#577" class="Generalizable">b</a> <a id="579" href="Algebra.Morphism.html#579" class="Generalizable">ℓ₁</a> <a id="582" href="Algebra.Morphism.html#582" class="Generalizable">ℓ₂</a> <a id="585" class="Symbol">:</a> <a id="587" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="597" href="Algebra.Morphism.html#597" class="Generalizable">A</a> <a id="599" class="Symbol">:</a> <a id="601" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="605" href="Algebra.Morphism.html#575" class="Generalizable">a</a>
<a id="611" href="Algebra.Morphism.html#611" class="Generalizable">B</a> <a id="613" class="Symbol">:</a> <a id="615" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="619" href="Algebra.Morphism.html#577" class="Generalizable">b</a>
<a id="622" class="Comment">------------------------------------------------------------------------</a>
<a id="695" class="Comment">-- Re-export</a>
<a id="709" class="Keyword">module</a> <a id="Definitions"></a><a id="716" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="728" class="Symbol">{</a><a id="729" href="Algebra.Morphism.html#729" class="Bound">a</a> <a id="731" href="Algebra.Morphism.html#731" class="Bound">b</a> <a id="733" href="Algebra.Morphism.html#733" class="Bound">ℓ₁</a><a id="735" class="Symbol">}</a> <a id="737" class="Symbol">(</a><a id="738" href="Algebra.Morphism.html#738" class="Bound">A</a> <a id="740" class="Symbol">:</a> <a id="742" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="746" href="Algebra.Morphism.html#729" class="Bound">a</a><a id="747" class="Symbol">)</a> <a id="749" class="Symbol">(</a><a id="750" href="Algebra.Morphism.html#750" class="Bound">B</a> <a id="752" class="Symbol">:</a> <a id="754" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="758" href="Algebra.Morphism.html#731" class="Bound">b</a><a id="759" class="Symbol">)</a> <a id="761" class="Symbol">(</a><a id="762" href="Algebra.Morphism.html#762" class="Bound Operator">_≈_</a> <a id="766" class="Symbol">:</a> <a id="768" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="772" href="Algebra.Morphism.html#750" class="Bound">B</a> <a id="774" href="Algebra.Morphism.html#733" class="Bound">ℓ₁</a><a id="776" class="Symbol">)</a> <a id="778" class="Keyword">where</a>
<a id="786" class="Keyword">open</a> <a id="791" href="Algebra.Morphism.Definitions.html" class="Module">MorphismDefinitions</a> <a id="811" href="Algebra.Morphism.html#738" class="Bound">A</a> <a id="813" href="Algebra.Morphism.html#750" class="Bound">B</a> <a id="815" href="Algebra.Morphism.html#762" class="Bound Operator">_≈_</a> <a id="819" class="Keyword">public</a>
<a id="827" class="Keyword">open</a> <a id="832" class="Keyword">import</a> <a id="839" href="Algebra.Morphism.Structures.html" class="Module">Algebra.Morphism.Structures</a> <a id="867" class="Keyword">public</a>
<a id="876" class="Comment">------------------------------------------------------------------------</a>
<a id="949" class="Comment">-- DEPRECATED</a>
<a id="963" class="Comment">------------------------------------------------------------------------</a>
<a id="1036" class="Comment">-- Please use the new definitions re-exported from</a>
<a id="1087" class="Comment">-- `Algebra.Morphism.Structures` as continuing support for the below is</a>
<a id="1159" class="Comment">-- no guaranteed.</a>
<a id="1178" class="Comment">-- Version 1.5</a>
<a id="1194" class="Keyword">module</a> <a id="1201" href="Algebra.Morphism.html#1201" class="Module">_</a> <a id="1203" class="Symbol">{</a><a id="1204" href="Algebra.Morphism.html#1204" class="Bound">c₁</a> <a id="1207" href="Algebra.Morphism.html#1207" class="Bound">ℓ₁</a> <a id="1210" href="Algebra.Morphism.html#1210" class="Bound">c₂</a> <a id="1213" href="Algebra.Morphism.html#1213" class="Bound">ℓ₂</a><a id="1215" class="Symbol">}</a>
<a id="1226" class="Symbol">(</a><a id="1227" href="Algebra.Morphism.html#1227" class="Bound">From</a> <a id="1232" class="Symbol">:</a> <a id="1234" href="Algebra.Bundles.html#2152" class="Record">Semigroup</a> <a id="1244" href="Algebra.Morphism.html#1204" class="Bound">c₁</a> <a id="1247" href="Algebra.Morphism.html#1207" class="Bound">ℓ₁</a><a id="1249" class="Symbol">)</a>
<a id="1260" class="Symbol">(</a><a id="1261" href="Algebra.Morphism.html#1261" class="Bound">To</a> <a id="1266" class="Symbol">:</a> <a id="1268" href="Algebra.Bundles.html#2152" class="Record">Semigroup</a> <a id="1278" href="Algebra.Morphism.html#1210" class="Bound">c₂</a> <a id="1281" href="Algebra.Morphism.html#1213" class="Bound">ℓ₂</a><a id="1283" class="Symbol">)</a> <a id="1285" class="Keyword">where</a>
<a id="1294" class="Keyword">private</a>
<a id="1306" class="Keyword">module</a> <a id="1313" href="Algebra.Morphism.html#1313" class="Module">F</a> <a id="1315" class="Symbol">=</a> <a id="1317" href="Algebra.Bundles.html#2152" class="Module">Semigroup</a> <a id="1327" href="Algebra.Morphism.html#1227" class="Bound">From</a>
<a id="1336" class="Keyword">module</a> <a id="1343" href="Algebra.Morphism.html#1343" class="Module">T</a> <a id="1345" class="Symbol">=</a> <a id="1347" href="Algebra.Bundles.html#2152" class="Module">Semigroup</a> <a id="1357" href="Algebra.Morphism.html#1261" class="Bound">To</a>
<a id="1362" class="Keyword">open</a> <a id="1367" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="1379" href="Algebra.Bundles.html#2234" class="Function">F.Carrier</a> <a id="1389" href="Algebra.Bundles.html#2234" class="Field">T.Carrier</a> <a id="1399" href="Algebra.Bundles.html#2258" class="Field Operator">T._≈_</a>
<a id="1408" class="Keyword">record</a> <a id="1415" href="Algebra.Morphism.html#1415" class="Record">IsSemigroupMorphism</a> <a id="1435" class="Symbol">(</a><a id="1436" href="Algebra.Morphism.html#1436" class="Bound Operator">⟦_⟧</a> <a id="1440" class="Symbol">:</a> <a id="1442" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="1450" class="Symbol">)</a> <a id="1452" class="Symbol">:</a>
<a id="1463" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1467" class="Symbol">(</a><a id="1468" href="Algebra.Morphism.html#1204" class="Bound">c₁</a> <a id="1471" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1473" href="Algebra.Morphism.html#1207" class="Bound">ℓ₁</a> <a id="1476" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1478" href="Algebra.Morphism.html#1210" class="Bound">c₂</a> <a id="1481" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1483" href="Algebra.Morphism.html#1213" class="Bound">ℓ₂</a><a id="1485" class="Symbol">)</a> <a id="1487" class="Keyword">where</a>
<a id="1497" class="Keyword">field</a>
<a id="1509" href="Algebra.Morphism.html#1509" class="Field">⟦⟧-cong</a> <a id="1517" class="Symbol">:</a> <a id="1519" href="Algebra.Morphism.html#1436" class="Bound Operator">⟦_⟧</a> <a id="1523" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="1533" href="Algebra.Bundles.html#2258" class="Function Operator">F._≈_</a> <a id="1539" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="1541" href="Algebra.Bundles.html#2258" class="Field Operator">T._≈_</a>
<a id="1553" href="Algebra.Morphism.html#1553" class="Field">∙-homo</a> <a id="1561" class="Symbol">:</a> <a id="1563" href="Algebra.Morphism.Definitions.html#852" class="Function">Homomorphic₂</a> <a id="1576" href="Algebra.Morphism.html#1436" class="Bound Operator">⟦_⟧</a> <a id="1580" href="Algebra.Bundles.html#2290" class="Function Operator">F._∙_</a> <a id="1586" href="Algebra.Bundles.html#2290" class="Field Operator">T._∙_</a>
<a id="1595" href="Algebra.Morphism.html#1595" class="Function">IsSemigroupMorphism-syntax</a> <a id="1622" class="Symbol">=</a> <a id="1624" href="Algebra.Morphism.html#1415" class="Record">IsSemigroupMorphism</a>
<a id="1646" class="Keyword">syntax</a> <a id="1653" href="Algebra.Morphism.html#1595" class="Function">IsSemigroupMorphism-syntax</a> <a id="1680" class="Bound">From</a> <a id="1685" class="Bound">To</a> <a id="1688" class="Bound">F</a> <a id="1690" class="Symbol">=</a> <a id="1692" class="Bound">F</a> <a id="1694" class="Function">Is</a> <a id="1697" class="Bound">From</a> <a id="1702" class="Function">-Semigroup⟶</a> <a id="1714" class="Bound">To</a>
<a id="1718" class="Keyword">module</a> <a id="1725" href="Algebra.Morphism.html#1725" class="Module">_</a> <a id="1727" class="Symbol">{</a><a id="1728" href="Algebra.Morphism.html#1728" class="Bound">c₁</a> <a id="1731" href="Algebra.Morphism.html#1731" class="Bound">ℓ₁</a> <a id="1734" href="Algebra.Morphism.html#1734" class="Bound">c₂</a> <a id="1737" href="Algebra.Morphism.html#1737" class="Bound">ℓ₂</a><a id="1739" class="Symbol">}</a>
<a id="1750" class="Symbol">(</a><a id="1751" href="Algebra.Morphism.html#1751" class="Bound">From</a> <a id="1756" class="Symbol">:</a> <a id="1758" href="Algebra.Bundles.html#4430" class="Record">Monoid</a> <a id="1765" href="Algebra.Morphism.html#1728" class="Bound">c₁</a> <a id="1768" href="Algebra.Morphism.html#1731" class="Bound">ℓ₁</a><a id="1770" class="Symbol">)</a>
<a id="1781" class="Symbol">(</a><a id="1782" href="Algebra.Morphism.html#1782" class="Bound">To</a> <a id="1787" class="Symbol">:</a> <a id="1789" href="Algebra.Bundles.html#4430" class="Record">Monoid</a> <a id="1796" href="Algebra.Morphism.html#1734" class="Bound">c₂</a> <a id="1799" href="Algebra.Morphism.html#1737" class="Bound">ℓ₂</a><a id="1801" class="Symbol">)</a> <a id="1803" class="Keyword">where</a>
<a id="1812" class="Keyword">private</a>
<a id="1824" class="Keyword">module</a> <a id="1831" href="Algebra.Morphism.html#1831" class="Module">F</a> <a id="1833" class="Symbol">=</a> <a id="1835" href="Algebra.Bundles.html#4430" class="Module">Monoid</a> <a id="1842" href="Algebra.Morphism.html#1751" class="Bound">From</a>
<a id="1851" class="Keyword">module</a> <a id="1858" href="Algebra.Morphism.html#1858" class="Module">T</a> <a id="1860" class="Symbol">=</a> <a id="1862" href="Algebra.Bundles.html#4430" class="Module">Monoid</a> <a id="1869" href="Algebra.Morphism.html#1782" class="Bound">To</a>
<a id="1874" class="Keyword">open</a> <a id="1879" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="1891" href="Algebra.Bundles.html#4509" class="Function">F.Carrier</a> <a id="1901" href="Algebra.Bundles.html#4509" class="Field">T.Carrier</a> <a id="1911" href="Algebra.Bundles.html#4530" class="Field Operator">T._≈_</a>
<a id="1920" class="Keyword">record</a> <a id="1927" href="Algebra.Morphism.html#1927" class="Record">IsMonoidMorphism</a> <a id="1944" class="Symbol">(</a><a id="1945" href="Algebra.Morphism.html#1945" class="Bound Operator">⟦_⟧</a> <a id="1949" class="Symbol">:</a> <a id="1951" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="1959" class="Symbol">)</a> <a id="1961" class="Symbol">:</a>
<a id="1972" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1976" class="Symbol">(</a><a id="1977" href="Algebra.Morphism.html#1728" class="Bound">c₁</a> <a id="1980" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1982" href="Algebra.Morphism.html#1731" class="Bound">ℓ₁</a> <a id="1985" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1987" href="Algebra.Morphism.html#1734" class="Bound">c₂</a> <a id="1990" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1992" href="Algebra.Morphism.html#1737" class="Bound">ℓ₂</a><a id="1994" class="Symbol">)</a> <a id="1996" class="Keyword">where</a>
<a id="2006" class="Keyword">field</a>
<a id="2018" href="Algebra.Morphism.html#2018" class="Field">sm-homo</a> <a id="2026" class="Symbol">:</a> <a id="2028" href="Algebra.Morphism.html#1415" class="Record">IsSemigroupMorphism</a> <a id="2048" href="Algebra.Bundles.html#4675" class="Function">F.semigroup</a> <a id="2060" href="Algebra.Bundles.html#4675" class="Function">T.semigroup</a> <a id="2072" href="Algebra.Morphism.html#1945" class="Bound Operator">⟦_⟧</a>
<a id="2082" href="Algebra.Morphism.html#2082" class="Field">ε-homo</a> <a id="2090" class="Symbol">:</a> <a id="2092" href="Algebra.Morphism.Definitions.html#680" class="Function">Homomorphic₀</a> <a id="2105" href="Algebra.Morphism.html#1945" class="Bound Operator">⟦_⟧</a> <a id="2109" href="Algebra.Bundles.html#4586" class="Function">F.ε</a> <a id="2113" href="Algebra.Bundles.html#4586" class="Field">T.ε</a>
<a id="2122" class="Keyword">open</a> <a id="2127" href="Algebra.Morphism.html#1415" class="Module">IsSemigroupMorphism</a> <a id="2147" href="Algebra.Morphism.html#2018" class="Field">sm-homo</a> <a id="2155" class="Keyword">public</a>
<a id="2165" href="Algebra.Morphism.html#2165" class="Function">IsMonoidMorphism-syntax</a> <a id="2189" class="Symbol">=</a> <a id="2191" href="Algebra.Morphism.html#1927" class="Record">IsMonoidMorphism</a>
<a id="2210" class="Keyword">syntax</a> <a id="2217" href="Algebra.Morphism.html#2165" class="Function">IsMonoidMorphism-syntax</a> <a id="2241" class="Bound">From</a> <a id="2246" class="Bound">To</a> <a id="2249" class="Bound">F</a> <a id="2251" class="Symbol">=</a> <a id="2253" class="Bound">F</a> <a id="2255" class="Function">Is</a> <a id="2258" class="Bound">From</a> <a id="2263" class="Function">-Monoid⟶</a> <a id="2272" class="Bound">To</a>
<a id="2276" class="Keyword">module</a> <a id="2283" href="Algebra.Morphism.html#2283" class="Module">_</a> <a id="2285" class="Symbol">{</a><a id="2286" href="Algebra.Morphism.html#2286" class="Bound">c₁</a> <a id="2289" href="Algebra.Morphism.html#2289" class="Bound">ℓ₁</a> <a id="2292" href="Algebra.Morphism.html#2292" class="Bound">c₂</a> <a id="2295" href="Algebra.Morphism.html#2295" class="Bound">ℓ₂</a><a id="2297" class="Symbol">}</a>
<a id="2308" class="Symbol">(</a><a id="2309" href="Algebra.Morphism.html#2309" class="Bound">From</a> <a id="2314" class="Symbol">:</a> <a id="2316" href="Algebra.Bundles.html#4910" class="Record">CommutativeMonoid</a> <a id="2334" href="Algebra.Morphism.html#2286" class="Bound">c₁</a> <a id="2337" href="Algebra.Morphism.html#2289" class="Bound">ℓ₁</a><a id="2339" class="Symbol">)</a>
<a id="2350" class="Symbol">(</a><a id="2351" href="Algebra.Morphism.html#2351" class="Bound">To</a> <a id="2356" class="Symbol">:</a> <a id="2358" href="Algebra.Bundles.html#4910" class="Record">CommutativeMonoid</a> <a id="2376" href="Algebra.Morphism.html#2292" class="Bound">c₂</a> <a id="2379" href="Algebra.Morphism.html#2295" class="Bound">ℓ₂</a><a id="2381" class="Symbol">)</a> <a id="2383" class="Keyword">where</a>
<a id="2392" class="Keyword">private</a>
<a id="2404" class="Keyword">module</a> <a id="2411" href="Algebra.Morphism.html#2411" class="Module">F</a> <a id="2413" class="Symbol">=</a> <a id="2415" href="Algebra.Bundles.html#4910" class="Module">CommutativeMonoid</a> <a id="2433" href="Algebra.Morphism.html#2309" class="Bound">From</a>
<a id="2442" class="Keyword">module</a> <a id="2449" href="Algebra.Morphism.html#2449" class="Module">T</a> <a id="2451" class="Symbol">=</a> <a id="2453" href="Algebra.Bundles.html#4910" class="Module">CommutativeMonoid</a> <a id="2471" href="Algebra.Morphism.html#2351" class="Bound">To</a>
<a id="2476" class="Keyword">open</a> <a id="2481" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="2493" href="Algebra.Bundles.html#5000" class="Function">F.Carrier</a> <a id="2503" href="Algebra.Bundles.html#5000" class="Field">T.Carrier</a> <a id="2513" href="Algebra.Bundles.html#5032" class="Field Operator">T._≈_</a>
<a id="2522" class="Keyword">record</a> <a id="2529" href="Algebra.Morphism.html#2529" class="Record">IsCommutativeMonoidMorphism</a> <a id="2557" class="Symbol">(</a><a id="2558" href="Algebra.Morphism.html#2558" class="Bound Operator">⟦_⟧</a> <a id="2562" class="Symbol">:</a> <a id="2564" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="2572" class="Symbol">)</a> <a id="2574" class="Symbol">:</a>
<a id="2585" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2589" class="Symbol">(</a><a id="2590" href="Algebra.Morphism.html#2286" class="Bound">c₁</a> <a id="2593" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2595" href="Algebra.Morphism.html#2289" class="Bound">ℓ₁</a> <a id="2598" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2600" href="Algebra.Morphism.html#2292" class="Bound">c₂</a> <a id="2603" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2605" href="Algebra.Morphism.html#2295" class="Bound">ℓ₂</a><a id="2607" class="Symbol">)</a> <a id="2609" class="Keyword">where</a>
<a id="2619" class="Keyword">field</a>
<a id="2631" href="Algebra.Morphism.html#2631" class="Field">mn-homo</a> <a id="2639" class="Symbol">:</a> <a id="2641" href="Algebra.Morphism.html#1927" class="Record">IsMonoidMorphism</a> <a id="2658" href="Algebra.Bundles.html#5254" class="Function">F.monoid</a> <a id="2667" href="Algebra.Bundles.html#5254" class="Function">T.monoid</a> <a id="2676" href="Algebra.Morphism.html#2558" class="Bound Operator">⟦_⟧</a>
<a id="2685" class="Keyword">open</a> <a id="2690" href="Algebra.Morphism.html#1927" class="Module">IsMonoidMorphism</a> <a id="2707" href="Algebra.Morphism.html#2631" class="Field">mn-homo</a> <a id="2715" class="Keyword">public</a>
<a id="2725" href="Algebra.Morphism.html#2725" class="Function">IsCommutativeMonoidMorphism-syntax</a> <a id="2760" class="Symbol">=</a> <a id="2762" href="Algebra.Morphism.html#2529" class="Record">IsCommutativeMonoidMorphism</a>
<a id="2792" class="Keyword">syntax</a> <a id="2799" href="Algebra.Morphism.html#2725" class="Function">IsCommutativeMonoidMorphism-syntax</a> <a id="2834" class="Bound">From</a> <a id="2839" class="Bound">To</a> <a id="2842" class="Bound">F</a> <a id="2844" class="Symbol">=</a> <a id="2846" class="Bound">F</a> <a id="2848" class="Function">Is</a> <a id="2851" class="Bound">From</a> <a id="2856" class="Function">-CommutativeMonoid⟶</a> <a id="2876" class="Bound">To</a>
<a id="2880" class="Keyword">module</a> <a id="2887" href="Algebra.Morphism.html#2887" class="Module">_</a> <a id="2889" class="Symbol">{</a><a id="2890" href="Algebra.Morphism.html#2890" class="Bound">c₁</a> <a id="2893" href="Algebra.Morphism.html#2893" class="Bound">ℓ₁</a> <a id="2896" href="Algebra.Morphism.html#2896" class="Bound">c₂</a> <a id="2899" href="Algebra.Morphism.html#2899" class="Bound">ℓ₂</a><a id="2901" class="Symbol">}</a>
<a id="2912" class="Symbol">(</a><a id="2913" href="Algebra.Morphism.html#2913" class="Bound">From</a> <a id="2918" class="Symbol">:</a> <a id="2920" href="Algebra.Bundles.html#5630" class="Record">IdempotentCommutativeMonoid</a> <a id="2948" href="Algebra.Morphism.html#2890" class="Bound">c₁</a> <a id="2951" href="Algebra.Morphism.html#2893" class="Bound">ℓ₁</a><a id="2953" class="Symbol">)</a>
<a id="2964" class="Symbol">(</a><a id="2965" href="Algebra.Morphism.html#2965" class="Bound">To</a> <a id="2970" class="Symbol">:</a> <a id="2972" href="Algebra.Bundles.html#5630" class="Record">IdempotentCommutativeMonoid</a> <a id="3000" href="Algebra.Morphism.html#2896" class="Bound">c₂</a> <a id="3003" href="Algebra.Morphism.html#2899" class="Bound">ℓ₂</a><a id="3005" class="Symbol">)</a> <a id="3007" class="Keyword">where</a>
<a id="3016" class="Keyword">private</a>
<a id="3028" class="Keyword">module</a> <a id="3035" href="Algebra.Morphism.html#3035" class="Module">F</a> <a id="3037" class="Symbol">=</a> <a id="3039" href="Algebra.Bundles.html#5630" class="Module">IdempotentCommutativeMonoid</a> <a id="3067" href="Algebra.Morphism.html#2913" class="Bound">From</a>
<a id="3076" class="Keyword">module</a> <a id="3083" href="Algebra.Morphism.html#3083" class="Module">T</a> <a id="3085" class="Symbol">=</a> <a id="3087" href="Algebra.Bundles.html#5630" class="Module">IdempotentCommutativeMonoid</a> <a id="3115" href="Algebra.Morphism.html#2965" class="Bound">To</a>
<a id="3120" class="Keyword">open</a> <a id="3125" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="3137" href="Algebra.Bundles.html#5730" class="Function">F.Carrier</a> <a id="3147" href="Algebra.Bundles.html#5730" class="Field">T.Carrier</a> <a id="3157" href="Algebra.Bundles.html#5772" class="Field Operator">T._≈_</a>
<a id="3166" class="Keyword">record</a> <a id="3173" href="Algebra.Morphism.html#3173" class="Record">IsIdempotentCommutativeMonoidMorphism</a> <a id="3211" class="Symbol">(</a><a id="3212" href="Algebra.Morphism.html#3212" class="Bound Operator">⟦_⟧</a> <a id="3216" class="Symbol">:</a> <a id="3218" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="3226" class="Symbol">)</a> <a id="3228" class="Symbol">:</a>
<a id="3239" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3243" class="Symbol">(</a><a id="3244" href="Algebra.Morphism.html#2890" class="Bound">c₁</a> <a id="3247" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="3249" href="Algebra.Morphism.html#2893" class="Bound">ℓ₁</a> <a id="3252" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="3254" href="Algebra.Morphism.html#2896" class="Bound">c₂</a> <a id="3257" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="3259" href="Algebra.Morphism.html#2899" class="Bound">ℓ₂</a><a id="3261" class="Symbol">)</a> <a id="3263" class="Keyword">where</a>
<a id="3273" class="Keyword">field</a>
<a id="3285" href="Algebra.Morphism.html#3285" class="Field">mn-homo</a> <a id="3293" class="Symbol">:</a> <a id="3295" href="Algebra.Morphism.html#1927" class="Record">IsMonoidMorphism</a> <a id="3312" href="Algebra.Bundles.html#5254" class="Function">F.monoid</a> <a id="3321" href="Algebra.Bundles.html#5254" class="Function">T.monoid</a> <a id="3330" href="Algebra.Morphism.html#3212" class="Bound Operator">⟦_⟧</a>
<a id="3339" class="Keyword">open</a> <a id="3344" href="Algebra.Morphism.html#1927" class="Module">IsMonoidMorphism</a> <a id="3361" href="Algebra.Morphism.html#3285" class="Field">mn-homo</a> <a id="3369" class="Keyword">public</a>
<a id="3381" href="Algebra.Morphism.html#3381" class="Function">isCommutativeMonoidMorphism</a> <a id="3409" class="Symbol">:</a>
<a id="3417" href="Algebra.Morphism.html#2529" class="Record">IsCommutativeMonoidMorphism</a> <a id="3445" href="Algebra.Bundles.html#6064" class="Function">F.commutativeMonoid</a> <a id="3465" href="Algebra.Bundles.html#6064" class="Function">T.commutativeMonoid</a> <a id="3485" href="Algebra.Morphism.html#3212" class="Bound Operator">⟦_⟧</a>
<a id="3493" href="Algebra.Morphism.html#3381" class="Function">isCommutativeMonoidMorphism</a> <a id="3521" class="Symbol">=</a> <a id="3523" class="Keyword">record</a> <a id="3530" class="Symbol">{</a> <a id="3532" href="Algebra.Morphism.html#2631" class="Field">mn-homo</a> <a id="3540" class="Symbol">=</a> <a id="3542" href="Algebra.Morphism.html#3285" class="Field">mn-homo</a> <a id="3550" class="Symbol">}</a>
<a id="3555" href="Algebra.Morphism.html#3555" class="Function">IsIdempotentCommutativeMonoidMorphism-syntax</a> <a id="3600" class="Symbol">=</a> <a id="3602" href="Algebra.Morphism.html#3173" class="Record">IsIdempotentCommutativeMonoidMorphism</a>
<a id="3642" class="Keyword">syntax</a> <a id="3649" href="Algebra.Morphism.html#3555" class="Function">IsIdempotentCommutativeMonoidMorphism-syntax</a> <a id="3694" class="Bound">From</a> <a id="3699" class="Bound">To</a> <a id="3702" class="Bound">F</a> <a id="3704" class="Symbol">=</a> <a id="3706" class="Bound">F</a> <a id="3708" class="Function">Is</a> <a id="3711" class="Bound">From</a> <a id="3716" class="Function">-IdempotentCommutativeMonoid⟶</a> <a id="3746" class="Bound">To</a>
<a id="3750" class="Keyword">module</a> <a id="3757" href="Algebra.Morphism.html#3757" class="Module">_</a> <a id="3759" class="Symbol">{</a><a id="3760" href="Algebra.Morphism.html#3760" class="Bound">c₁</a> <a id="3763" href="Algebra.Morphism.html#3763" class="Bound">ℓ₁</a> <a id="3766" href="Algebra.Morphism.html#3766" class="Bound">c₂</a> <a id="3769" href="Algebra.Morphism.html#3769" class="Bound">ℓ₂</a><a id="3771" class="Symbol">}</a>
<a id="3782" class="Symbol">(</a><a id="3783" href="Algebra.Morphism.html#3783" class="Bound">From</a> <a id="3788" class="Symbol">:</a> <a id="3790" href="Algebra.Bundles.html#7318" class="Record">Group</a> <a id="3796" href="Algebra.Morphism.html#3760" class="Bound">c₁</a> <a id="3799" href="Algebra.Morphism.html#3763" class="Bound">ℓ₁</a><a id="3801" class="Symbol">)</a>
<a id="3812" class="Symbol">(</a><a id="3813" href="Algebra.Morphism.html#3813" class="Bound">To</a> <a id="3818" class="Symbol">:</a> <a id="3820" href="Algebra.Bundles.html#7318" class="Record">Group</a> <a id="3826" href="Algebra.Morphism.html#3766" class="Bound">c₂</a> <a id="3829" href="Algebra.Morphism.html#3769" class="Bound">ℓ₂</a><a id="3831" class="Symbol">)</a> <a id="3833" class="Keyword">where</a>
<a id="3842" class="Keyword">private</a>
<a id="3854" class="Keyword">module</a> <a id="3861" href="Algebra.Morphism.html#3861" class="Module">F</a> <a id="3863" class="Symbol">=</a> <a id="3865" href="Algebra.Bundles.html#7318" class="Module">Group</a> <a id="3871" href="Algebra.Morphism.html#3783" class="Bound">From</a>
<a id="3880" class="Keyword">module</a> <a id="3887" href="Algebra.Morphism.html#3887" class="Module">T</a> <a id="3889" class="Symbol">=</a> <a id="3891" href="Algebra.Bundles.html#7318" class="Module">Group</a> <a id="3897" href="Algebra.Morphism.html#3813" class="Bound">To</a>
<a id="3902" class="Keyword">open</a> <a id="3907" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="3919" href="Algebra.Bundles.html#7411" class="Function">F.Carrier</a> <a id="3929" href="Algebra.Bundles.html#7411" class="Field">T.Carrier</a> <a id="3939" href="Algebra.Bundles.html#7431" class="Field Operator">T._≈_</a>
<a id="3948" class="Keyword">record</a> <a id="3955" href="Algebra.Morphism.html#3955" class="Record">IsGroupMorphism</a> <a id="3971" class="Symbol">(</a><a id="3972" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦_⟧</a> <a id="3976" class="Symbol">:</a> <a id="3978" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="3986" class="Symbol">)</a> <a id="3988" class="Symbol">:</a>
<a id="3999" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="4003" class="Symbol">(</a><a id="4004" href="Algebra.Morphism.html#3760" class="Bound">c₁</a> <a id="4007" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4009" href="Algebra.Morphism.html#3763" class="Bound">ℓ₁</a> <a id="4012" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4014" href="Algebra.Morphism.html#3766" class="Bound">c₂</a> <a id="4017" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4019" href="Algebra.Morphism.html#3769" class="Bound">ℓ₂</a><a id="4021" class="Symbol">)</a> <a id="4023" class="Keyword">where</a>
<a id="4033" class="Keyword">field</a>
<a id="4045" href="Algebra.Morphism.html#4045" class="Field">mn-homo</a> <a id="4053" class="Symbol">:</a> <a id="4055" href="Algebra.Morphism.html#1927" class="Record">IsMonoidMorphism</a> <a id="4072" href="Algebra.Bundles.html#7688" class="Function">F.monoid</a> <a id="4081" href="Algebra.Bundles.html#7688" class="Function">T.monoid</a> <a id="4090" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦_⟧</a>
<a id="4099" class="Keyword">open</a> <a id="4104" href="Algebra.Morphism.html#1927" class="Module">IsMonoidMorphism</a> <a id="4121" href="Algebra.Morphism.html#4045" class="Field">mn-homo</a> <a id="4129" class="Keyword">public</a>
<a id="4141" href="Algebra.Morphism.html#4141" class="Function">⁻¹-homo</a> <a id="4149" class="Symbol">:</a> <a id="4151" href="Algebra.Morphism.Definitions.html#753" class="Function">Homomorphic₁</a> <a id="4164" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦_⟧</a> <a id="4168" href="Algebra.Bundles.html#7507" class="Function Operator">F._⁻¹</a> <a id="4174" href="Algebra.Bundles.html#7507" class="Field Operator">T._⁻¹</a>
<a id="4184" href="Algebra.Morphism.html#4141" class="Function">⁻¹-homo</a> <a id="4192" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4194" class="Symbol">=</a> <a id="4196" class="Keyword">let</a> <a id="4200" class="Keyword">open</a> <a id="4205" href="Relation.Binary.Reasoning.Setoid.html" class="Module">EqR</a> <a id="4209" href="Algebra.Structures.html#1294" class="Function">T.setoid</a> <a id="4218" class="Keyword">in</a> <a id="4221" href="Algebra.Structures.html#4812" class="Function">T.uniqueˡ-⁻¹</a> <a id="4234" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦</a> <a id="4236" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4238" href="Algebra.Bundles.html#7507" class="Function Operator">F.⁻¹</a> <a id="4243" href="Algebra.Morphism.html#3972" class="Bound Operator">⟧</a> <a id="4245" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦</a> <a id="4247" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4249" href="Algebra.Morphism.html#3972" class="Bound Operator">⟧</a> <a id="4251" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="4253" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="4265" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦</a> <a id="4267" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4269" href="Algebra.Bundles.html#7507" class="Function Operator">F.⁻¹</a> <a id="4274" href="Algebra.Morphism.html#3972" class="Bound Operator">⟧</a> <a id="4276" href="Algebra.Bundles.html#7459" class="Field Operator">T.∙</a> <a id="4280" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦</a> <a id="4282" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4284" href="Algebra.Morphism.html#3972" class="Bound Operator">⟧</a> <a id="4286" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4289" href="Relation.Binary.Structures.html#1603" class="Function">T.sym</a> <a id="4295" class="Symbol">(</a><a id="4296" href="Algebra.Morphism.html#1553" class="Function">∙-homo</a> <a id="4303" class="Symbol">(</a><a id="4304" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4306" href="Algebra.Bundles.html#7507" class="Function Operator">F.⁻¹</a><a id="4310" class="Symbol">)</a> <a id="4312" href="Algebra.Morphism.html#4192" class="Bound">x</a><a id="4313" class="Symbol">)</a> <a id="4315" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4323" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦</a> <a id="4325" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4327" href="Algebra.Bundles.html#7507" class="Function Operator">F.⁻¹</a> <a id="4332" href="Algebra.Bundles.html#7459" class="Function Operator">F.∙</a> <a id="4336" href="Algebra.Morphism.html#4192" class="Bound">x</a> <a id="4338" href="Algebra.Morphism.html#3972" class="Bound Operator">⟧</a> <a id="4344" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4347" href="Algebra.Morphism.html#1509" class="Function">⟦⟧-cong</a> <a id="4355" class="Symbol">(</a><a id="4356" href="Algebra.Structures.html#4685" class="Function">F.inverseˡ</a> <a id="4367" href="Algebra.Morphism.html#4192" class="Bound">x</a><a id="4368" class="Symbol">)</a> <a id="4370" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4378" href="Algebra.Morphism.html#3972" class="Bound Operator">⟦</a> <a id="4380" href="Algebra.Bundles.html#7485" class="Function">F.ε</a> <a id="4384" href="Algebra.Morphism.html#3972" class="Bound Operator">⟧</a> <a id="4399" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4402" href="Algebra.Morphism.html#2082" class="Function">ε-homo</a> <a id="4409" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4417" href="Algebra.Bundles.html#7485" class="Field">T.ε</a> <a id="4421" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="4426" href="Algebra.Morphism.html#4426" class="Function">IsGroupMorphism-syntax</a> <a id="4449" class="Symbol">=</a> <a id="4451" href="Algebra.Morphism.html#3955" class="Record">IsGroupMorphism</a>
<a id="4469" class="Keyword">syntax</a> <a id="4476" href="Algebra.Morphism.html#4426" class="Function">IsGroupMorphism-syntax</a> <a id="4499" class="Bound">From</a> <a id="4504" class="Bound">To</a> <a id="4507" class="Bound">F</a> <a id="4509" class="Symbol">=</a> <a id="4511" class="Bound">F</a> <a id="4513" class="Function">Is</a> <a id="4516" class="Bound">From</a> <a id="4521" class="Function">-Group⟶</a> <a id="4529" class="Bound">To</a>
<a id="4533" class="Keyword">module</a> <a id="4540" href="Algebra.Morphism.html#4540" class="Module">_</a> <a id="4542" class="Symbol">{</a><a id="4543" href="Algebra.Morphism.html#4543" class="Bound">c₁</a> <a id="4546" href="Algebra.Morphism.html#4546" class="Bound">ℓ₁</a> <a id="4549" href="Algebra.Morphism.html#4549" class="Bound">c₂</a> <a id="4552" href="Algebra.Morphism.html#4552" class="Bound">ℓ₂</a><a id="4554" class="Symbol">}</a>
<a id="4565" class="Symbol">(</a><a id="4566" href="Algebra.Morphism.html#4566" class="Bound">From</a> <a id="4571" class="Symbol">:</a> <a id="4573" href="Algebra.Bundles.html#7842" class="Record">AbelianGroup</a> <a id="4586" href="Algebra.Morphism.html#4543" class="Bound">c₁</a> <a id="4589" href="Algebra.Morphism.html#4546" class="Bound">ℓ₁</a><a id="4591" class="Symbol">)</a>
<a id="4602" class="Symbol">(</a><a id="4603" href="Algebra.Morphism.html#4603" class="Bound">To</a> <a id="4608" class="Symbol">:</a> <a id="4610" href="Algebra.Bundles.html#7842" class="Record">AbelianGroup</a> <a id="4623" href="Algebra.Morphism.html#4549" class="Bound">c₂</a> <a id="4626" href="Algebra.Morphism.html#4552" class="Bound">ℓ₂</a><a id="4628" class="Symbol">)</a> <a id="4630" class="Keyword">where</a>
<a id="4639" class="Keyword">private</a>
<a id="4651" class="Keyword">module</a> <a id="4658" href="Algebra.Morphism.html#4658" class="Module">F</a> <a id="4660" class="Symbol">=</a> <a id="4662" href="Algebra.Bundles.html#7842" class="Module">AbelianGroup</a> <a id="4675" href="Algebra.Morphism.html#4566" class="Bound">From</a>
<a id="4684" class="Keyword">module</a> <a id="4691" href="Algebra.Morphism.html#4691" class="Module">T</a> <a id="4693" class="Symbol">=</a> <a id="4695" href="Algebra.Bundles.html#7842" class="Module">AbelianGroup</a> <a id="4708" href="Algebra.Morphism.html#4603" class="Bound">To</a>
<a id="4713" class="Keyword">open</a> <a id="4718" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="4730" href="Algebra.Bundles.html#7942" class="Function">F.Carrier</a> <a id="4740" href="Algebra.Bundles.html#7942" class="Field">T.Carrier</a> <a id="4750" href="Algebra.Bundles.html#7969" class="Field Operator">T._≈_</a>
<a id="4759" class="Keyword">record</a> <a id="4766" href="Algebra.Morphism.html#4766" class="Record">IsAbelianGroupMorphism</a> <a id="4789" class="Symbol">(</a><a id="4790" href="Algebra.Morphism.html#4790" class="Bound Operator">⟦_⟧</a> <a id="4794" class="Symbol">:</a> <a id="4796" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="4804" class="Symbol">)</a> <a id="4806" class="Symbol">:</a>
<a id="4817" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="4821" class="Symbol">(</a><a id="4822" href="Algebra.Morphism.html#4543" class="Bound">c₁</a> <a id="4825" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4827" href="Algebra.Morphism.html#4546" class="Bound">ℓ₁</a> <a id="4830" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4832" href="Algebra.Morphism.html#4549" class="Bound">c₂</a> <a id="4835" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4837" href="Algebra.Morphism.html#4552" class="Bound">ℓ₂</a><a id="4839" class="Symbol">)</a> <a id="4841" class="Keyword">where</a>
<a id="4851" class="Keyword">field</a>
<a id="4863" href="Algebra.Morphism.html#4863" class="Field">gp-homo</a> <a id="4871" class="Symbol">:</a> <a id="4873" href="Algebra.Morphism.html#3955" class="Record">IsGroupMorphism</a> <a id="4889" href="Algebra.Bundles.html#8193" class="Function">F.group</a> <a id="4897" href="Algebra.Bundles.html#8193" class="Function">T.group</a> <a id="4905" href="Algebra.Morphism.html#4790" class="Bound Operator">⟦_⟧</a>
<a id="4914" class="Keyword">open</a> <a id="4919" href="Algebra.Morphism.html#3955" class="Module">IsGroupMorphism</a> <a id="4935" href="Algebra.Morphism.html#4863" class="Field">gp-homo</a> <a id="4943" class="Keyword">public</a>
<a id="4953" href="Algebra.Morphism.html#4953" class="Function">IsAbelianGroupMorphism-syntax</a> <a id="4983" class="Symbol">=</a> <a id="4985" href="Algebra.Morphism.html#4766" class="Record">IsAbelianGroupMorphism</a>
<a id="5010" class="Keyword">syntax</a> <a id="5017" href="Algebra.Morphism.html#4953" class="Function">IsAbelianGroupMorphism-syntax</a> <a id="5047" class="Bound">From</a> <a id="5052" class="Bound">To</a> <a id="5055" class="Bound">F</a> <a id="5057" class="Symbol">=</a> <a id="5059" class="Bound">F</a> <a id="5061" class="Function">Is</a> <a id="5064" class="Bound">From</a> <a id="5069" class="Function">-AbelianGroup⟶</a> <a id="5084" class="Bound">To</a>
<a id="5088" class="Keyword">module</a> <a id="5095" href="Algebra.Morphism.html#5095" class="Module">_</a> <a id="5097" class="Symbol">{</a><a id="5098" href="Algebra.Morphism.html#5098" class="Bound">c₁</a> <a id="5101" href="Algebra.Morphism.html#5101" class="Bound">ℓ₁</a> <a id="5104" href="Algebra.Morphism.html#5104" class="Bound">c₂</a> <a id="5107" href="Algebra.Morphism.html#5107" class="Bound">ℓ₂</a><a id="5109" class="Symbol">}</a>
<a id="5120" class="Symbol">(</a><a id="5121" href="Algebra.Morphism.html#5121" class="Bound">From</a> <a id="5126" class="Symbol">:</a> <a id="5128" href="Algebra.Bundles.html#20872" class="Record">Ring</a> <a id="5133" href="Algebra.Morphism.html#5098" class="Bound">c₁</a> <a id="5136" href="Algebra.Morphism.html#5101" class="Bound">ℓ₁</a><a id="5138" class="Symbol">)</a>
<a id="5149" class="Symbol">(</a><a id="5150" href="Algebra.Morphism.html#5150" class="Bound">To</a> <a id="5155" class="Symbol">:</a> <a id="5157" href="Algebra.Bundles.html#20872" class="Record">Ring</a> <a id="5162" href="Algebra.Morphism.html#5104" class="Bound">c₂</a> <a id="5165" href="Algebra.Morphism.html#5107" class="Bound">ℓ₂</a><a id="5167" class="Symbol">)</a> <a id="5169" class="Keyword">where</a>
<a id="5178" class="Keyword">private</a>
<a id="5190" class="Keyword">module</a> <a id="5197" href="Algebra.Morphism.html#5197" class="Module">F</a> <a id="5199" class="Symbol">=</a> <a id="5201" href="Algebra.Bundles.html#20872" class="Module">Ring</a> <a id="5206" href="Algebra.Morphism.html#5121" class="Bound">From</a>
<a id="5215" class="Keyword">module</a> <a id="5222" href="Algebra.Morphism.html#5222" class="Module">T</a> <a id="5224" class="Symbol">=</a> <a id="5226" href="Algebra.Bundles.html#20872" class="Module">Ring</a> <a id="5231" href="Algebra.Morphism.html#5150" class="Bound">To</a>
<a id="5236" class="Keyword">open</a> <a id="5241" href="Algebra.Morphism.html#716" class="Module">Definitions</a> <a id="5253" href="Algebra.Bundles.html#20978" class="Function">F.Carrier</a> <a id="5263" href="Algebra.Bundles.html#20978" class="Field">T.Carrier</a> <a id="5273" href="Algebra.Bundles.html#20998" class="Field Operator">T._≈_</a>
<a id="5282" class="Keyword">record</a> <a id="5289" href="Algebra.Morphism.html#5289" class="Record">IsRingMorphism</a> <a id="5304" class="Symbol">(</a><a id="5305" href="Algebra.Morphism.html#5305" class="Bound Operator">⟦_⟧</a> <a id="5309" class="Symbol">:</a> <a id="5311" href="Algebra.Morphism.Definitions.html#1236" class="Function">Morphism</a><a id="5319" class="Symbol">)</a> <a id="5321" class="Symbol">:</a>
<a id="5332" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="5336" class="Symbol">(</a><a id="5337" href="Algebra.Morphism.html#5098" class="Bound">c₁</a> <a id="5340" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="5342" href="Algebra.Morphism.html#5101" class="Bound">ℓ₁</a> <a id="5345" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="5347" href="Algebra.Morphism.html#5104" class="Bound">c₂</a> <a id="5350" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="5352" href="Algebra.Morphism.html#5107" class="Bound">ℓ₂</a><a id="5354" class="Symbol">)</a> <a id="5356" class="Keyword">where</a>
<a id="5366" class="Keyword">field</a>
<a id="5378" href="Algebra.Morphism.html#5378" class="Field">+-abgp-homo</a> <a id="5390" class="Symbol">:</a> <a id="5392" href="Algebra.Morphism.html#5305" class="Bound Operator">⟦_⟧</a> <a id="5396" href="Algebra.Morphism.html#4953" class="Function">Is</a> <a id="5399" href="Algebra.Bundles.html#21218" class="Function">F.+-abelianGroup</a> <a id="5416" href="Algebra.Morphism.html#4953" class="Function">-AbelianGroup⟶</a> <a id="5431" href="Algebra.Bundles.html#21218" class="Function">T.+-abelianGroup</a>
<a id="5454" href="Algebra.Morphism.html#5454" class="Field">*-mn-homo</a> <a id="5466" class="Symbol">:</a> <a id="5468" href="Algebra.Morphism.html#5305" class="Bound Operator">⟦_⟧</a> <a id="5472" href="Algebra.Morphism.html#2165" class="Function">Is</a> <a id="5475" href="Algebra.Bundles.html#15998" class="Function">F.*-monoid</a> <a id="5486" href="Algebra.Morphism.html#2165" class="Function">-Monoid⟶</a> <a id="5495" href="Algebra.Bundles.html#15998" class="Function">T.*-monoid</a>
<a id="5509" href="Algebra.Morphism.html#5509" class="Function">IsRingMorphism-syntax</a> <a id="5531" class="Symbol">=</a> <a id="5533" href="Algebra.Morphism.html#5289" class="Record">IsRingMorphism</a>
<a id="5550" class="Keyword">syntax</a> <a id="5557" href="Algebra.Morphism.html#5509" class="Function">IsRingMorphism-syntax</a> <a id="5579" class="Bound">From</a> <a id="5584" class="Bound">To</a> <a id="5587" class="Bound">F</a> <a id="5589" class="Symbol">=</a> <a id="5591" class="Bound">F</a> <a id="5593" class="Function">Is</a> <a id="5596" class="Bound">From</a> <a id="5601" class="Function">-Ring⟶</a> <a id="5608" class="Bound">To</a>
<a id="5612" class="Symbol">{-#</a> <a id="5616" class="Keyword">WARNING_ON_USAGE</a> <a id="5633" class="Pragma">IsSemigroupMorphism</a>
<a id="5653" class="String">&quot;Warning: IsSemigroupMorphism was deprecated in v1.5.
Please use IsSemigroupHomomorphism instead.&quot;</a>
<a id="5752" class="Symbol">#-}</a>
<a id="5756" class="Symbol">{-#</a> <a id="5760" class="Keyword">WARNING_ON_USAGE</a> <a id="5777" class="Pragma">IsMonoidMorphism</a>
<a id="5794" class="String">&quot;Warning: IsMonoidMorphism was deprecated in v1.5.
Please use IsMonoidHomomorphism instead.&quot;</a>
<a id="5887" class="Symbol">#-}</a>
<a id="5891" class="Symbol">{-#</a> <a id="5895" class="Keyword">WARNING_ON_USAGE</a> <a id="5912" class="Pragma">IsCommutativeMonoidMorphism</a>
<a id="5940" class="String">&quot;Warning: IsCommutativeMonoidMorphism was deprecated in v1.5.
Please use IsMonoidHomomorphism instead.&quot;</a>
<a id="6044" class="Symbol">#-}</a>
<a id="6048" class="Symbol">{-#</a> <a id="6052" class="Keyword">WARNING_ON_USAGE</a> <a id="6069" class="Pragma">IsIdempotentCommutativeMonoidMorphism</a>
<a id="6107" class="String">&quot;Warning: IsIdempotentCommutativeMonoidMorphism was deprecated in v1.5.
Please use IsMonoidHomomorphism instead.&quot;</a>
<a id="6221" class="Symbol">#-}</a>
<a id="6225" class="Symbol">{-#</a> <a id="6229" class="Keyword">WARNING_ON_USAGE</a> <a id="6246" class="Pragma">IsGroupMorphism</a>
<a id="6262" class="String">&quot;Warning: IsGroupMorphism was deprecated in v1.5.
Please use IsGroupHomomorphism instead.&quot;</a>
<a id="6353" class="Symbol">#-}</a>
<a id="6357" class="Symbol">{-#</a> <a id="6361" class="Keyword">WARNING_ON_USAGE</a> <a id="6378" class="Pragma">IsAbelianGroupMorphism</a>
<a id="6401" class="String">&quot;Warning: IsAbelianGroupMorphism was deprecated in v1.5.
Please use IsGroupHomomorphism instead.&quot;</a>
<a id="6499" class="Symbol">#-}</a>
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