Return-path: Envelope-to: jbovlaste-admin@lojban.org Delivery-date: Fri, 12 Aug 2022 10:55:40 -0700 Received: from [192.168.123.254] (port=59542 helo=jiten.lojban.org) by d7893716a6e6 with smtp (Exim 4.94.2) (envelope-from ) id 1oMYsj-001Ptw-9m for jbovlaste-admin@lojban.org; Fri, 12 Aug 2022 10:55:40 -0700 Received: by jiten.lojban.org (sSMTP sendmail emulation); Fri, 12 Aug 2022 17:55:37 +0000 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word facni -- By krtisfranks Date: Fri, 12 Aug 2022 17:55:37 +0000 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Message-Id: X-Spam-Score: -2.9 (--) X-Spam_score: -2.9 X-Spam_score_int: -28 X-Spam_bar: -- In jbovlaste, the user krtisfranks has edited a definition of "facni" in the language "English". Differences: 5,5c5,5 < $x_2$ and $x_3$ cannot merely be sets; they must be structures which each are a set endowed with at least one operator each; the ith operator endowing one space corresponds to exactly the ith operator endowing the other space under mapping $x_1$. For any operator of $x_2$, $x_1$ is commutative with it with respect to functional composition ({fa'ai}) when the (other) operator is 'translated' to the corresponding operator of $x_3$ appropriately. $x_1$ is linear/a linear operator; $x_1$ is a homomorphism; $x_1$ distributes. $x_2$ is homomorphic with $x_3$ under $x_1$; they need not be identical (in fact, their respective operators need not even be identical, just 'homomorphically similar'). For "distributivity"/"distributive property", "linearity of operator", or "homomorphicity of operator", use "ka(m)( )facni" with $x_1$ filled with "{ce'u}"; for "homomorphicity of spaces", use the same thing, but with $x_2$ or $x_3$ filled with "{ce'u}". See also: {socni}, {cajni}, {sezni}, {dukni}; {fa'ai}; {fatri}. This is a structure-operator-preserving function, and thus is an example of a {stodraunju}. --- > $x_2$ and $x_3$ cannot merely be sets; they must be structures/systems which each are a set/category endowed with at least one operator/relation/property (here, "operator" will refer to any of these options) each; the $i$th operator endowing one space corresponds to exactly the ith operator endowing the other space under mapping $x_1$. For any operator of $x_2$, $x_1$ is commutative with it with respect to functional composition ({fa'ai}) when the (other) operator is 'translated' to the corresponding operator of $x_3$ appropriately. $x_1$ is linear/a linear operator; $x_1$ is a homomorphism; $x_1$ distributes. $x_2$ is homomorphic with $x_3$ under $x_1$; they need not be identical (in fact, their respective operators need not even be identical, just 'homomorphically similar'). For "distributivity"/"distributive property", "linearity of operator", or "homomorphicity of operator", use "ka(m)( )facni" with $x_1$ filled with "{ce'u}"; for "homomorphicity of spaces", use the same thing, but with $x_2$ or $x_3$ filled with "{ce'u}". See also: "{socni}", "{cajni}", "{sezni}", "{dukni}"; "{fa'ai}"; "{fatri}". This is a structure-operator-preserving function, and thus is an example of a {stodraunju}. 17a18,19 \n> Word: linear map, In Sense: vector space homomorphism > Word: graph homomorphism, In Sense: Old Data: Definition: $x_1$ is an n-ary operator/map which is distributive/linear/homomorphic in or over or from space/structure $x_2$, mapping to space or structure $x_3$, thereby producing a new space/structure $x_4$ which is the 'union' of $x_2$ and $x_3$ endowed with $x_1$; $x_1$ distributes over/through all of the operators of $x_2$. Notes: $x_2$ and $x_3$ cannot merely be sets; they must be structures which each are a set endowed with at least one operator each; the ith operator endowing one space corresponds to exactly the ith operator endowing the other space under mapping $x_1$. For any operator of $x_2$, $x_1$ is commutative with it with respect to functional composition ({fa'ai}) when the (other) operator is 'translated' to the corresponding operator of $x_3$ appropriately. $x_1$ is linear/a linear operator; $x_1$ is a homomorphism; $x_1$ distributes. $x_2$ is homomorphic with $x_3$ under $x_1$; they need not be identical (in fact, their respective operators need not even be identical, just 'homomorphically similar'). For "distributivity"/"distributive property", "linearity of operator", or "homomorphicity of operator", use "ka(m)( )facni" with $x_1$ filled with "{ce'u}"; for "homomorphicity of spaces", use the same thing, but with $x_2$ or $x_3$ filled with "{ce'u}". See also: {socni}, {cajni}, {sezni}, {dukni}; {fa'ai}; {fatri}. This is a structure-operator-preserving function, and thus is an example of a {stodraunju}. Jargon: Gloss Keywords: Word: distributive function, In Sense: Word: distributive operator, In Sense: Word: distributive property, In Sense: Word: homomorphicity, In Sense: Word: homomorphism, In Sense: algebra Word: linear function, In Sense: linear algebra Word: linear operator, In Sense: Place Keywords: New Data: Definition: $x_1$ is an n-ary operator/map which is distributive/linear/homomorphic in or over or from space/structure $x_2$, mapping to space or structure $x_3$, thereby producing a new space/structure $x_4$ which is the 'union' of $x_2$ and $x_3$ endowed with $x_1$; $x_1$ distributes over/through all of the operators of $x_2$. Notes: $x_2$ and $x_3$ cannot merely be sets; they must be structures/systems which each are a set/category endowed with at least one operator/relation/property (here, "operator" will refer to any of these options) each; the $i$th operator endowing one space corresponds to exactly the ith operator endowing the other space under mapping $x_1$. For any operator of $x_2$, $x_1$ is commutative with it with respect to functional composition ({fa'ai}) when the (other) operator is 'translated' to the corresponding operator of $x_3$ appropriately. $x_1$ is linear/a linear operator; $x_1$ is a homomorphism; $x_1$ distributes. $x_2$ is homomorphic with $x_3$ under $x_1$; they need not be identical (in fact, their respective operators need not even be identical, just 'homomorphically similar'). For "distributivity"/"distributive property", "linearity of operator", or "homomorphicity of operator", use "ka(m)( )facni" with $x_1$ filled with "{ce'u}"; for "homomorphicity of spaces", use the same thing, but with $x_2$ or $x_3$ filled with "{ce'u}". See also: "{socni}", "{cajni}", "{sezni}", "{dukni}"; "{fa'ai}"; "{fatri}". This is a structure-operator-preserving function, and thus is an example of a {stodraunju}. Jargon: Gloss Keywords: Word: distributive function, In Sense: Word: distributive operator, In Sense: Word: distributive property, In Sense: Word: homomorphicity, In Sense: Word: homomorphism, In Sense: algebra Word: linear function, In Sense: linear algebra Word: linear operator, In Sense: Word: linear map, In Sense: vector space homomorphism Word: graph homomorphism, In Sense: Place Keywords: You can go to to see it.