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Type | Label | Description |
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Statement | ||
Theorem | xpsmnd0 18701 | The identity element of a binary product of monoids. (Contributed by AV, 25-Feb-2025.) |
β’ π = (π Γs π) β β’ ((π β Mnd β§ π β Mnd) β (0gβπ) = β¨(0gβπ ), (0gβπ)β©) | ||
Theorem | mnd1 18702 | The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.) |
β’ π = {β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©} β β’ (πΌ β π β π β Mnd) | ||
Theorem | mnd1id 18703 | The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.) |
β’ π = {β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©} β β’ (πΌ β π β (0gβπ) = πΌ) | ||
Syntax | cmhm 18704 | Hom-set generator class for monoids. |
class MndHom | ||
Syntax | csubmnd 18705 | Class function taking a monoid to its lattice of submonoids. |
class SubMnd | ||
Definition | df-mhm 18706* | A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ MndHom = (π β Mnd, π‘ β Mnd β¦ {π β ((Baseβπ‘) βm (Baseβπ )) β£ (βπ₯ β (Baseβπ )βπ¦ β (Baseβπ )(πβ(π₯(+gβπ )π¦)) = ((πβπ₯)(+gβπ‘)(πβπ¦)) β§ (πβ(0gβπ )) = (0gβπ‘))}) | ||
Definition | df-submnd 18707* | A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ SubMnd = (π β Mnd β¦ {π‘ β π« (Baseβπ ) β£ ((0gβπ ) β π‘ β§ βπ₯ β π‘ βπ¦ β π‘ (π₯(+gβπ )π¦) β π‘)}) | ||
Theorem | ismhm 18708* | Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (0gβπ) β β’ (πΉ β (π MndHom π) β ((π β Mnd β§ π β Mnd) β§ (πΉ:π΅βΆπΆ β§ βπ₯ β π΅ βπ¦ β π΅ (πΉβ(π₯ + π¦)) = ((πΉβπ₯) ⨣ (πΉβπ¦)) β§ (πΉβ 0 ) = π))) | ||
Theorem | ismhmd 18709* | Deduction version of ismhm 18708. (Contributed by SN, 27-Jul-2024.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (0gβπ) & β’ (π β π β Mnd) & β’ (π β π β Mnd) & β’ (π β πΉ:π΅βΆπΆ) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (πΉβ(π₯ + π¦)) = ((πΉβπ₯) ⨣ (πΉβπ¦))) & β’ (π β (πΉβ 0 ) = π) β β’ (π β πΉ β (π MndHom π)) | ||
Theorem | mhmrcl1 18710 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ (πΉ β (π MndHom π) β π β Mnd) | ||
Theorem | mhmrcl2 18711 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ (πΉ β (π MndHom π) β π β Mnd) | ||
Theorem | mhmf 18712 | A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) β β’ (πΉ β (π MndHom π) β πΉ:π΅βΆπΆ) | ||
Theorem | ismhm0 18713 | Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (0gβπ) β β’ (πΉ β (π MndHom π) β ((π β Mnd β§ π β Mnd) β§ (πΉ β (π MgmHom π) β§ (πΉβ 0 ) = π))) | ||
Theorem | mhmismgmhm 18714 | Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.) |
β’ (πΉ β (π MndHom π) β πΉ β (π MgmHom π)) | ||
Theorem | mhmpropd 18715* | Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.) |
β’ (π β π΅ = (Baseβπ½)) & β’ (π β πΆ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ (π β πΆ = (Baseβπ)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπ½)π¦) = (π₯(+gβπΏ)π¦)) & β’ ((π β§ (π₯ β πΆ β§ π¦ β πΆ)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπ)π¦)) β β’ (π β (π½ MndHom πΎ) = (πΏ MndHom π)) | ||
Theorem | mhmlin 18716 | A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) β β’ ((πΉ β (π MndHom π) β§ π β π΅ β§ π β π΅) β (πΉβ(π + π)) = ((πΉβπ) ⨣ (πΉβπ))) | ||
Theorem | mhm0 18717 | A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ 0 = (0gβπ) & β’ π = (0gβπ) β β’ (πΉ β (π MndHom π) β (πΉβ 0 ) = π) | ||
Theorem | idmhm 18718 | The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.) |
β’ π΅ = (Baseβπ) β β’ (π β Mnd β ( I βΎ π΅) β (π MndHom π)) | ||
Theorem | mhmf1o 18719 | A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.) |
β’ π΅ = (Baseβπ ) & β’ πΆ = (Baseβπ) β β’ (πΉ β (π MndHom π) β (πΉ:π΅β1-1-ontoβπΆ β β‘πΉ β (π MndHom π ))) | ||
Theorem | submrcl 18720 | Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ (π β (SubMndβπ) β π β Mnd) | ||
Theorem | issubm 18721* | Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ) & β’ + = (+gβπ) β β’ (π β Mnd β (π β (SubMndβπ) β (π β π΅ β§ 0 β π β§ βπ₯ β π βπ¦ β π (π₯ + π¦) β π))) | ||
Theorem | issubm2 18722 | Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ) & β’ π» = (π βΎs π) β β’ (π β Mnd β (π β (SubMndβπ) β (π β π΅ β§ 0 β π β§ π» β Mnd))) | ||
Theorem | issubmndb 18723 | The submonoid predicate. Analogous to issubg 19043. (Contributed by AV, 1-Feb-2024.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) β β’ (π β (SubMndβπΊ) β ((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π))) | ||
Theorem | issubmd 18724* | Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & β’ 0 = (0gβπ) & β’ (π β π β Mnd) & β’ (π β π) & β’ ((π β§ ((π₯ β π΅ β§ π¦ β π΅) β§ (π β§ π))) β π) & β’ (π§ = 0 β (π β π)) & β’ (π§ = π₯ β (π β π)) & β’ (π§ = π¦ β (π β π)) & β’ (π§ = (π₯ + π¦) β (π β π)) β β’ (π β {π§ β π΅ β£ π} β (SubMndβπ)) | ||
Theorem | mndissubm 18725 | If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. Analogous to grpissubg 19063. (Contributed by AV, 17-Feb-2024.) |
β’ π΅ = (BaseβπΊ) & β’ π = (Baseβπ») & β’ 0 = (0gβπΊ) β β’ ((πΊ β Mnd β§ π» β Mnd) β ((π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π))) β π β (SubMndβπΊ))) | ||
Theorem | resmndismnd 18726 | If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the other monoid is contained in the base set of the monoid, then the other monoid restricted to the base set of the monoid is a monoid. Analogous to resgrpisgrp 19064. (Contributed by AV, 17-Feb-2024.) |
β’ π΅ = (BaseβπΊ) & β’ π = (Baseβπ») & β’ 0 = (0gβπΊ) β β’ ((πΊ β Mnd β§ π» β Mnd) β ((π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π))) β (πΊ βΎs π) β Mnd)) | ||
Theorem | submss 18727 | Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π΅ = (Baseβπ) β β’ (π β (SubMndβπ) β π β π΅) | ||
Theorem | submid 18728 | Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
β’ π΅ = (Baseβπ) β β’ (π β Mnd β π΅ β (SubMndβπ)) | ||
Theorem | subm0cl 18729 | Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ 0 = (0gβπ) β β’ (π β (SubMndβπ) β 0 β π) | ||
Theorem | submcl 18730 | Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.) |
β’ + = (+gβπ) β β’ ((π β (SubMndβπ) β§ π β π β§ π β π) β (π + π) β π) | ||
Theorem | submmnd 18731 | Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π» = (π βΎs π) β β’ (π β (SubMndβπ) β π» β Mnd) | ||
Theorem | submbas 18732 | The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.) |
β’ π» = (π βΎs π) β β’ (π β (SubMndβπ) β π = (Baseβπ»)) | ||
Theorem | subm0 18733 | Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
β’ π» = (π βΎs π) & β’ 0 = (0gβπ) β β’ (π β (SubMndβπ) β 0 = (0gβπ»)) | ||
Theorem | subsubm 18734 | A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.) |
β’ π» = (πΊ βΎs π) β β’ (π β (SubMndβπΊ) β (π΄ β (SubMndβπ») β (π΄ β (SubMndβπΊ) β§ π΄ β π))) | ||
Theorem | 0subm 18735 | The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.) |
β’ 0 = (0gβπΊ) β β’ (πΊ β Mnd β { 0 } β (SubMndβπΊ)) | ||
Theorem | insubm 18736 | The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.) |
β’ ((π΄ β (SubMndβπ) β§ π΅ β (SubMndβπ)) β (π΄ β© π΅) β (SubMndβπ)) | ||
Theorem | 0mhm 18737 | The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
β’ 0 = (0gβπ) & β’ π΅ = (Baseβπ) β β’ ((π β Mnd β§ π β Mnd) β (π΅ Γ { 0 }) β (π MndHom π)) | ||
Theorem | resmhm 18738 | Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.) |
β’ π = (π βΎs π) β β’ ((πΉ β (π MndHom π) β§ π β (SubMndβπ)) β (πΉ βΎ π) β (π MndHom π)) | ||
Theorem | resmhm2 18739 | One direction of resmhm2b 18740. (Contributed by Mario Carneiro, 18-Jun-2015.) |
β’ π = (π βΎs π) β β’ ((πΉ β (π MndHom π) β§ π β (SubMndβπ)) β πΉ β (π MndHom π)) | ||
Theorem | resmhm2b 18740 | Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.) |
β’ π = (π βΎs π) β β’ ((π β (SubMndβπ) β§ ran πΉ β π) β (πΉ β (π MndHom π) β πΉ β (π MndHom π))) | ||
Theorem | mhmco 18741 | The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
β’ ((πΉ β (π MndHom π) β§ πΊ β (π MndHom π)) β (πΉ β πΊ) β (π MndHom π)) | ||
Theorem | mhmimalem 18742* | Lemma for mhmima 18743 and similar theorems, formerly part of proof for mhmima 18743. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.) |
β’ (π β πΉ β (π MndHom π)) & β’ (π β π β (Baseβπ)) & β’ (π β β = (+gβπ)) & β’ (π β + = (+gβπ)) & β’ ((π β§ π§ β π β§ π₯ β π) β (π§ β π₯) β π) β β’ (π β βπ₯ β (πΉ β π)βπ¦ β (πΉ β π)(π₯ + π¦) β (πΉ β π)) | ||
Theorem | mhmima 18743 | The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.) (Proof shortened by AV, 8-Mar-2025.) |
β’ ((πΉ β (π MndHom π) β§ π β (SubMndβπ)) β (πΉ β π) β (SubMndβπ)) | ||
Theorem | mhmeql 18744 | The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
β’ ((πΉ β (π MndHom π) β§ πΊ β (π MndHom π)) β dom (πΉ β© πΊ) β (SubMndβπ)) | ||
Theorem | submacs 18745 | Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ π΅ = (BaseβπΊ) β β’ (πΊ β Mnd β (SubMndβπΊ) β (ACSβπ΅)) | ||
Theorem | mndind 18746* | Induction in a monoid. In this theorem, π(π₯) is the "generic" proposition to be be proved (the first four hypotheses tell its values at y, y+z, 0, A respectively). The two induction hypotheses mndind.i1 and mndind.i2 tell that it is true at 0, that if it is true at y then it is true at y+z (provided z is in πΊ). The hypothesis mndind.k tells that πΊ is generating. (Contributed by SO, 14-Jul-2018.) |
β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + π§) β (π β π)) & β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π΄ β (π β π)) & β’ 0 = (0gβπ) & β’ + = (+gβπ) & β’ π΅ = (Baseβπ) & β’ (π β π β Mnd) & β’ (π β πΊ β π΅) & β’ (π β π΅ = ((mrClsβ(SubMndβπ))βπΊ)) & β’ (π β π) & β’ (((π β§ π¦ β π΅ β§ π§ β πΊ) β§ π) β π) & β’ (π β π΄ β π΅) β β’ (π β π) | ||
Theorem | prdspjmhm 18747* | A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.) |
β’ π = (πXsπ ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β π) & β’ (π β π :πΌβΆMnd) & β’ (π β π΄ β πΌ) β β’ (π β (π₯ β π΅ β¦ (π₯βπ΄)) β (π MndHom (π βπ΄))) | ||
Theorem | pwspjmhm 18748* | A projection from a structure power of a monoid to the monoid itself is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) β β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (π₯ β π΅ β¦ (π₯βπ΄)) β (π MndHom π )) | ||
Theorem | pwsdiagmhm 18749* | Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ πΉ = (π₯ β π΅ β¦ (πΌ Γ {π₯})) β β’ ((π β Mnd β§ πΌ β π) β πΉ β (π MndHom π)) | ||
Theorem | pwsco1mhm 18750* | Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
β’ π = (π βs π΄) & β’ π = (π βs π΅) & β’ πΆ = (Baseβπ) & β’ (π β π β Mnd) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΉ:π΄βΆπ΅) β β’ (π β (π β πΆ β¦ (π β πΉ)) β (π MndHom π)) | ||
Theorem | pwsco2mhm 18751* | Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
β’ π = (π βs π΄) & β’ π = (π βs π΄) & β’ π΅ = (Baseβπ) & β’ (π β π΄ β π) & β’ (π β πΉ β (π MndHom π)) β β’ (π β (π β π΅ β¦ (πΉ β π)) β (π MndHom π)) | ||
One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 17393. If order is not significant, it is simpler to use families instead. | ||
Theorem | gsumvallem2 18752* | Lemma for properties of the set of identities of πΊ. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ + = (+gβπΊ) & β’ π = {π₯ β π΅ β£ βπ¦ β π΅ ((π₯ + π¦) = π¦ β§ (π¦ + π₯) = π¦)} β β’ (πΊ β Mnd β π = { 0 }) | ||
Theorem | gsumsubm 18753 | Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.) |
β’ (π β π΄ β π) & β’ (π β π β (SubMndβπΊ)) & β’ (π β πΉ:π΄βΆπ) & β’ π» = (πΊ βΎs π) β β’ (π β (πΊ Ξ£g πΉ) = (π» Ξ£g πΉ)) | ||
Theorem | gsumz 18754* | Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
β’ 0 = (0gβπΊ) β β’ ((πΊ β Mnd β§ π΄ β π) β (πΊ Ξ£g (π β π΄ β¦ 0 )) = 0 ) | ||
Theorem | gsumwsubmcl 18755 | Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) |
β’ ((π β (SubMndβπΊ) β§ π β Word π) β (πΊ Ξ£g π) β π) | ||
Theorem | gsumws1 18756 | A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
β’ π΅ = (BaseβπΊ) β β’ (π β π΅ β (πΊ Ξ£g β¨βπββ©) = π) | ||
Theorem | gsumwcl 18757 | Closure of the composite of a word in a structure πΊ. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
β’ π΅ = (BaseβπΊ) β β’ ((πΊ β Mnd β§ π β Word π΅) β (πΊ Ξ£g π) β π΅) | ||
Theorem | gsumsgrpccat 18758 | Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 18759. (Contributed by AV, 26-Dec-2023.) |
β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((πΊ β Smgrp β§ (π β Word π΅ β§ π β Word π΅) β§ (π β β β§ π β β )) β (πΊ Ξ£g (π ++ π)) = ((πΊ Ξ£g π) + (πΊ Ξ£g π))) | ||
Theorem | gsumccat 18759 | Homomorphic property of composites. Second formula in [Lang] p. 4. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 26-Dec-2023.) |
β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((πΊ β Mnd β§ π β Word π΅ β§ π β Word π΅) β (πΊ Ξ£g (π ++ π)) = ((πΊ Ξ£g π) + (πΊ Ξ£g π))) | ||
Theorem | gsumws2 18760 | Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((πΊ β Mnd β§ π β π΅ β§ π β π΅) β (πΊ Ξ£g β¨βππββ©) = (π + π)) | ||
Theorem | gsumccatsn 18761 | Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.) |
β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((πΊ β Mnd β§ π β Word π΅ β§ π β π΅) β (πΊ Ξ£g (π ++ β¨βπββ©)) = ((πΊ Ξ£g π) + π)) | ||
Theorem | gsumspl 18762 | The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
β’ π΅ = (Baseβπ) & β’ (π β π β Mnd) & β’ (π β π β Word π΅) & β’ (π β πΉ β (0...π)) & β’ (π β π β (0...(β―βπ))) & β’ (π β π β Word π΅) & β’ (π β π β Word π΅) & β’ (π β (π Ξ£g π) = (π Ξ£g π)) β β’ (π β (π Ξ£g (π splice β¨πΉ, π, πβ©)) = (π Ξ£g (π splice β¨πΉ, π, πβ©))) | ||
Theorem | gsumwmhm 18763 | Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
β’ π΅ = (Baseβπ) β β’ ((π» β (π MndHom π) β§ π β Word π΅) β (π»β(π Ξ£g π)) = (π Ξ£g (π» β π))) | ||
Theorem | gsumwspan 18764* | The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ π΅ = (Baseβπ) & β’ πΎ = (mrClsβ(SubMndβπ)) β β’ ((π β Mnd β§ πΊ β π΅) β (πΎβπΊ) = ran (π€ β Word πΊ β¦ (π Ξ£g π€))) | ||
Syntax | cfrmd 18765 | Extend class definition with the free monoid construction. |
class freeMnd | ||
Syntax | cvrmd 18766 | Extend class notation with free monoid injection. |
class varFMnd | ||
Definition | df-frmd 18767 | Define a free monoid over a set π of generators, defined as the set of finite strings on πΌ with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ freeMnd = (π β V β¦ {β¨(Baseβndx), Word πβ©, β¨(+gβndx), ( ++ βΎ (Word π Γ Word π))β©}) | ||
Definition | df-vrmd 18768* | Define a free monoid over a set π of generators, defined as the set of finite strings on πΌ with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ varFMnd = (π β V β¦ (π β π β¦ β¨βπββ©)) | ||
Theorem | frmdval 18769 | Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ π = (freeMndβπΌ) & β’ (πΌ β π β π΅ = Word πΌ) & β’ + = ( ++ βΎ (π΅ Γ π΅)) β β’ (πΌ β π β π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©}) | ||
Theorem | frmdbas 18770 | The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (Baseβπ) β β’ (πΌ β π β π΅ = Word πΌ) | ||
Theorem | frmdelbas 18771 | An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (Baseβπ) β β’ (π β π΅ β π β Word πΌ) | ||
Theorem | frmdplusg 18772 | The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) (Proof shortened by AV, 6-Nov-2024.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (Baseβπ) & β’ + = (+gβπ) β β’ + = ( ++ βΎ (π΅ Γ π΅)) | ||
Theorem | frmdadd 18773 | Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (Baseβπ) & β’ + = (+gβπ) β β’ ((π β π΅ β§ π β π΅) β (π + π) = (π ++ π)) | ||
Theorem | vrmdfval 18774* | The canonical injection from the generating set πΌ to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
β’ π = (varFMndβπΌ) β β’ (πΌ β π β π = (π β πΌ β¦ β¨βπββ©)) | ||
Theorem | vrmdval 18775 | The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
β’ π = (varFMndβπΌ) β β’ ((πΌ β π β§ π΄ β πΌ) β (πβπ΄) = β¨βπ΄ββ©) | ||
Theorem | vrmdf 18776 | The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
β’ π = (varFMndβπΌ) β β’ (πΌ β π β π:πΌβΆWord πΌ) | ||
Theorem | frmdmnd 18777 | A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
β’ π = (freeMndβπΌ) β β’ (πΌ β π β π β Mnd) | ||
Theorem | frmd0 18778 | The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ π = (freeMndβπΌ) β β’ β = (0gβπ) | ||
Theorem | frmdsssubm 18779 | The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
β’ π = (freeMndβπΌ) β β’ ((πΌ β π β§ π½ β πΌ) β Word π½ β (SubMndβπ)) | ||
Theorem | frmdgsum 18780 | Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ π = (freeMndβπΌ) & β’ π = (varFMndβπΌ) β β’ ((πΌ β π β§ π β Word πΌ) β (π Ξ£g (π β π)) = π) | ||
Theorem | frmdss2 18781 | A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of π½ is Word π½". (Contributed by Mario Carneiro, 2-Oct-2015.) |
β’ π = (freeMndβπΌ) & β’ π = (varFMndβπΌ) β β’ ((πΌ β π β§ π½ β πΌ β§ π΄ β (SubMndβπ)) β ((π β π½) β π΄ β Word π½ β π΄)) | ||
Theorem | frmdup1 18782* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (BaseβπΊ) & β’ πΈ = (π₯ β Word πΌ β¦ (πΊ Ξ£g (π΄ β π₯))) & β’ (π β πΊ β Mnd) & β’ (π β πΌ β π) & β’ (π β π΄:πΌβΆπ΅) β β’ (π β πΈ β (π MndHom πΊ)) | ||
Theorem | frmdup2 18783* | The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (BaseβπΊ) & β’ πΈ = (π₯ β Word πΌ β¦ (πΊ Ξ£g (π΄ β π₯))) & β’ (π β πΊ β Mnd) & β’ (π β πΌ β π) & β’ (π β π΄:πΌβΆπ΅) & β’ π = (varFMndβπΌ) & β’ (π β π β πΌ) β β’ (π β (πΈβ(πβπ)) = (π΄βπ)) | ||
Theorem | frmdup3lem 18784* | Lemma for frmdup3 18785. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (BaseβπΊ) & β’ π = (varFMndβπΌ) β β’ (((πΊ β Mnd β§ πΌ β π β§ π΄:πΌβΆπ΅) β§ (πΉ β (π MndHom πΊ) β§ (πΉ β π) = π΄)) β πΉ = (π₯ β Word πΌ β¦ (πΊ Ξ£g (π΄ β π₯)))) | ||
Theorem | frmdup3 18785* | Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.) |
β’ π = (freeMndβπΌ) & β’ π΅ = (BaseβπΊ) & β’ π = (varFMndβπΌ) β β’ ((πΊ β Mnd β§ πΌ β π β§ π΄:πΌβΆπ΅) β β!π β (π MndHom πΊ)(π β π) = π΄) | ||
According to Wikipedia ("Endomorphism", 25-Jan-2024, https://en.wikipedia.org/wiki/Endomorphism) "An endofunction is a function whose domain is equal to its codomain.". An endofunction is sometimes also called "self-mapping" (see https://www.wikidata.org/wiki/Q1691962) or "self-map" (see https://mathworld.wolfram.com/Self-Map.html), in German "Selbstabbildung" (see https://de.wikipedia.org/wiki/Selbstabbildung). | ||
Syntax | cefmnd 18786 | Extend class notation to include the class of monoids of endofunctions. |
class EndoFMnd | ||
Definition | df-efmnd 18787* | Define the monoid of endofunctions on set π₯. We represent the monoid as the set of functions from π₯ to itself ((π₯ βm π₯)) under function composition, and topologize it as a function space assuming the set is discrete. Analogous to the former definition of SymGrp, see df-symg 19277 and symgvalstruct 19306. (Contributed by AV, 25-Jan-2024.) |
β’ EndoFMnd = (π₯ β V β¦ β¦(π₯ βm π₯) / πβ¦{β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π₯ Γ {π« π₯}))β©}) | ||
Theorem | efmnd 18788* | The monoid of endofunctions on set π΄. (Contributed by AV, 25-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (π΄ βm π΄) & β’ + = (π β π΅, π β π΅ β¦ (π β π)) & β’ π½ = (βtβ(π΄ Γ {π« π΄})) β β’ (π΄ β π β πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©}) | ||
Theorem | efmndbas 18789 | The base set of the monoid of endofunctions on class π΄. (Contributed by AV, 25-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ π΅ = (π΄ βm π΄) | ||
Theorem | efmndbasabf 18790* | The base set of the monoid of endofunctions on class π΄ is the set of functions from π΄ into itself. (Contributed by AV, 29-Mar-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ π΅ = {π β£ π:π΄βΆπ΄} | ||
Theorem | elefmndbas 18791 | Two ways of saying a function is a mapping of π΄ to itself. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β π β (πΉ β π΅ β πΉ:π΄βΆπ΄)) | ||
Theorem | elefmndbas2 18792 | Two ways of saying a function is a mapping of π΄ to itself. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 29-Mar-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (πΉ β π β (πΉ β π΅ β πΉ:π΄βΆπ΄)) | ||
Theorem | efmndbasf 18793 | Elements in the monoid of endofunctions on π΄ are functions from π΄ into itself. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (πΉ β π΅ β πΉ:π΄βΆπ΄) | ||
Theorem | efmndhash 18794 | The monoid of endofunctions on π objects has cardinality πβπ. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β Fin β (β―βπ΅) = ((β―βπ΄)β(β―βπ΄))) | ||
Theorem | efmndbasfi 18795 | The monoid of endofunctions on a finite set π΄ is finite. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β Fin β π΅ β Fin) | ||
Theorem | efmndfv 18796 | The function value of an endofunction. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) β β’ ((πΉ β π΅ β§ π β π΄) β (πΉβπ) β π΄) | ||
Theorem | efmndtset 18797 | The topology of the monoid of endofunctions on π΄. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) β β’ (π΄ β π β (βtβ(π΄ Γ {π« π΄})) = (TopSetβπΊ)) | ||
Theorem | efmndplusg 18798* | The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ + = (π β π΅, π β π΅ β¦ (π β π)) | ||
Theorem | efmndov 18799 | The value of the group operation of the monoid of endofunctions on π΄. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((π β π΅ β§ π β π΅) β (π + π) = (π β π)) | ||
Theorem | efmndcl 18800 | The group operation of the monoid of endofunctions on π΄ is closed. (Contributed by AV, 27-Jan-2024.) |
β’ πΊ = (EndoFMndβπ΄) & β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((π β π΅ β§ π β π΅) β (π + π) β π΅) |
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